# Stationary Coupled KdV Hierarchies
and Related Poisson Structures

Allan P. Fordy and Qing Huang School of Mathematics, University of Leeds,
Leeds LS2 9JT, UK. E-mail<EMAIL_ADDRESS>of Mathematics, Center
for Nonlinear Studies, Northwest University, Xi’an 710069, People’s Republic
of China E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we continue our analysis of the stationary flows of $M$
component, coupled KdV (cKdV) hierarchies and their modifications. We describe
the general structure of the $t_{1}$ and $t_{2}$ flows, using the case $M=3$
as our main example. One of our stationary reductions gives $N$ degrees of
freedom, superintegrable systems. When $N=1$ (for $t_{1}$) and $N=2$ (for
$t_{2}$), we have Poisson maps, which give multi-Hamiltonian representations
of the flows. We discuss the general structure of these Poisson tensors and
give explicit forms for the case $M=3$. In this case there are 3 modified
hierarchies, each with 4 Poisson brackets.

The stationary $t_{2}$ flow (for $N=2$) is separable in parabolic coordinates.
Each Poisson bracket has rank 4, with $M+1$ Casimirs. The $4\times 4$ “core”
of the Poisson tensors are nonsingular and related by a “recursion operator”.
The remaining part of each tensor is built out of the two commuting
Hamiltonian vector fields, depending upon the specific Casimirs. The Poisson
brackets are generalised to include the entire class of potential, separable
in parabolic coordinates. The Jacobi identity imposes specific dependence on
some parameters, representing the Casimirs of the extended canonical bracket.
This general case is no longer a stationary cKdV flow, with Lax
representation. We give a recursive procedure for constructing the Lax
representation of the stationary flow for all values of $M$, without having to
go through the stationary reduction.

Keywords: Coupled KdV, bi-Hamiltonian, stationary flow, Hamiltonian system,
Poisson matrix, super-integrability.

MSC: 35Q53, 37J37, 37K05, 70H06

##  1 Introduction

In this paper our main interest is the construction of integrable and
superintegrable finite dimensional Hamiltonian systems (see [16, 8, 15, 7])
with multi-Hamiltonian representations and possessing a Lax pair. One such
construction is to consider the stationary reductions of completely integrable
nonlinear evolution equations (PDEs of soliton type). The observation that the
stationary flows of the KdV hierarchy are themselves integrable, finite
dimensional Hamiltonian systems was made by Bogoyavlenskii and Novikov in 1976
[6]. These were later shown to be bi-Hamiltonian in [3] (see also [9]).
Stationary flows of other “soliton hierarchies” were discovered in [10, 4].

In [12] we revisited this subject and found bi- and tri-Hamiltonian
representations of a number of superintegrable systems. These were associated
with the first two nontrivial flows of the $M$ component cKdV hierarchies,
related to the energy-dependent Schrödinger equation (see [1, 2]). In [12] we
concentrated on the cases $M=1$ and $M=2$, corresponding to the KdV and
Dispersive Water Wave (DWW) hierarchies respectively. In one particular
coordinate system (associated with a squared eigenfunction substitution) these
flows share interesting common features, which we emphasise in this paper.
These features could already be seen in the KdV and DWW hierarchies, but are
further illustrated here for the $M=3$ case.

In particular, the stationary $t_{1}$ flow has a Hamiltonian function of the
form

$h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)+U\left({\bf
Q}^{2}\right),$ (1a) where ${\bf Q}^{2}=\sum_{1}^{N}Q_{i}^{2}$, and is
superintegrable, with a universal collection of first integrals. These systems
are known to be superintegrable (see [8] for the case $N=3$ and [5] in
general), but for the particular choices of $U\left(\sqrt{{\bf Q}^{2}}\right)$
which appear in this paper (which are specific polynomials of ${\bf Q}^{2}$),
they have a Lax formulation, as follows from being stationary reductions. For
the case $N=1$, we also give a multi-Hamiltonian formulation.

The stationary $t_{2}$ flow has a Hamiltonian function of the form

$h^{(Q)}=\frac{1}{2}P_{1}^{2}+\frac{1}{2}\sum_{i=2}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)+U\left(Q_{1},\sum_{i=2}^{N}Q_{i}^{2}\right),$
(1b)

which has the same universal set of integrals as (1a), but for $2\leq i<j\leq
N$, as well as an integral $f^{(Q)}$, which survives the reduction to the case
$N=2$. This case generalises the well known Hamiltonians, which are separable
in parabolic coordinates (see Equation 2.2.41 in [16]). The particular
potentials which arise as stationary flows belong to the polynomial sequence
and, as a consequence, these have a Lax formulation. Again, for the case
$N=2$, we also give a multi-Hamiltonian formulation. Furthermore, this multi-
Hamiltonian formulation can be extended to the entire class of potentials,
separable in parabolic coordinates.

In Section 2 we give a brief overview of cKdV hierarchies, giving some
additional details for the $M=3$ case. We also give some background on Miura
maps and some general constructions of stationary flows. In particular, we
derive two different representations of stationary flows, using the first and
second Hamiltonian structures to respectively define coordinates $\bf q$ and
$\bf Q$, which play an important role in the following sections.

In Section 3 we consider the stationary $t_{1}$ flow in 2 coordinate systems,
$\bf q$ (1 degree of freedom) and $\bf Q$ ($N$ degrees of freedom), deriving
the specific case of the Hamiltonian (1a) for $M=3$. For each case we derive
the Lax representation. When $N=1$, we derive a Poisson map, which gives a
second (non-canonical) Poisson bracket for each of the systems.

A similar approach to the stationary $t_{2}$ flow is presented in Section 4,
where now the $\bf q$ space has 2 degrees of freedom, with the $\bf Q$ space
still $N$ degrees of freedom. When $N=2$, we derive a Poisson map, which gives
a second (non-canonical) Poisson bracket for each of the systems.

Both of these stationary flows are actually quadri-Hamiltonian (for $M=3$),
but to understand this we need to consider the Miura maps of Section 2.1.1,
but in the context of the stationary flows. For $M=3$, we have 4 spaces,
labelled $u,w,v$ and $z$ (in the PDE case), giving us 4 stationary manifolds,
labelled $\bf q$, $\bf\hat{Q}$, $\bf\bar{Q}$ and $\bf\tilde{Q}$, which are
extended to include 3 additional parameters as dynamical variables. The latter
are related to Casimirs of the Hamiltonian operators of the original PDEs.
These $\bf q$ coordinates coincide with our previous $\bf q$, but the others
are different from $\bf Q$. On the stationary manifolds, the Miura maps give
rise to Poisson maps, leading to the construction of a sequences of 4 Poisson
brackets for each of these spaces. This construction is carried out for the
$t_{1}$ and $t_{2}$ flows in Sections 5 and 6 respectively. For the $\bf q$
space, the first two Poisson brackets coincide with the ones constructed in
Sections 3 and 4. We can then use the Poisson map between $\bf q$ and $\bf Q$
spaces to build a quadri-Hamiltonian formulation in the $\bf Q$ space. In
Section 6 we do this for the stationary $t_{2}$ flow (28), which is separable
in parabolic coordinates. We analyse the structure of the Poisson tensors and,
in Section 6.4.1, introduce a type of recursion operator connecting the core
parts of the tensors. The remaining part of the tensor is built in a specific
way from the two commuting functions $h^{(Q)}$ and $f^{(Q)}$.

In Section 7, we consider the general potential, separable in parabolic
coordinates (see Equation 2.2.41 in [16]). We consider a corresponding
generalisation of the 4 Poisson brackets found for the stationary $t_{2}$ flow
(28) in $\bf Q$ space. Remarkably, the only restriction on the general
potential is that they have a specific linear dependence on the Casimir
parameters mentioned above. However, imposing the Lax representation does
restrict the potential to the specific forms related to the stationary
reduction. We show that it is possible to build the Lax matrices in a
recursive way, avoiding the need to go through the lengthy process of the
stationary reduction.

##  2 Lax Representation of cKdV Equations

In [1, 2] a detailed analysis was given of cKdV equations, associated with
“energy dependent” Schrödinger operators. In [12] we gave a very brief review,
which we further reduce here.

This differential operator Lax pair for the $M$ component cKdV hierarchy is
conveniently rewritten in “zero-curvature” form:

$\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{x}=\left(\begin{array}[]{cc}0&1\\\ -u&0\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)=U\Psi,\;\;\;\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)_{t}=\left(\begin{array}[]{cc}-P_{x}&2P\\\\[2.84526pt]
-2uP-P_{xx}&P_{x}\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\\\[2.84526pt]
\psi_{2}\end{array}\right)=V\Psi,$ (2a) where
$u=\sum_{i=0}^{M-1}u_{i}\lambda^{i}-\lambda^{M}$. Consistency of this pair of
equations leads to $U_{t}-V_{x}+[U,V]=0\quad\Rightarrow\quad
u_{t}=(\partial_{x}^{3}+4u\partial_{x}+2u_{x})P.$ (2b) For a specific
hierarchy, we must substitute the correct form of $u$ (by choosing $M$).

The cKdV hierarchy is given by the polynomial expansion

$P^{(n)}=\sum_{k=0}^{n}\lambda^{n-k}P_{k},\quad\mbox{with}\quad P_{0}=1,$ (2c)

corresponding to $t=t_{n}$. The corresponding $V$ is called $V^{(n)}$, after
which (2a) takes the form

$U_{t_{n}}-V^{(n)}_{x}+\left[U,V^{(n)}\right]=0,\;\;\;\mbox{with}\;\;\;u_{t_{n}}=(\partial_{x}^{3}+4u\partial_{x}+2u_{x})P^{(n)}.$
(2d)

Equating coefficients of powers of $\lambda$ gives us a recursive formula for
$P_{k}$ and then formulae for $u_{it_{n}}$, which can be written in matrix
form to give the Hamiltonian formulation.

###  2.1 The Hierarchy when $M=3$

In [12] we considered the KdV ($M=1$) and DWW ($M=2$) cases. Here we use the
case $M=3$ to illustrate our results.

For the case $M=3$ we have
$u=u_{0}+u_{1}\lambda+u_{2}\lambda^{2}-\lambda^{3}$. The coefficients $P_{k}$
in (2c) are constructed recursively, the first few being

$\displaystyle P_{0}=1,\quad P_{1}=\frac{1}{2}u_{2},\quad
P_{2}=\frac{1}{8}(4u_{1}+3u_{2}^{2}),\quad
P_{3}=\frac{1}{16}(8u_{0}+12u_{1}u_{2}+5u_{2}^{3}),$ $\displaystyle
P_{4}=\frac{3}{4}u_{0}u_{2}+\frac{3}{8}u_{1}^{2}+\frac{15}{16}u_{1}u_{2}^{2}+\frac{35}{128}u_{2}^{4}+\frac{1}{8}u_{2xx},$
(3a) $\displaystyle
P_{5}=\frac{3}{4}u_{0}u_{1}+\frac{15}{16}u_{2}(u_{0}u_{2}+u_{1}^{2})+\frac{35}{32}u_{1}u_{2}^{3}+\frac{63}{256}u_{2}^{5}+\frac{1}{8}u_{1xx}+\frac{5}{16}u_{2}u_{2xx}+\frac{5}{32}u_{2x}^{2}.$

The general recursion, defined by (2d), is

$J_{0}P_{n-3}+J_{1}P_{n-2}+J_{2}P_{n-1}+J_{3}P_{n}=0,$ (3b)

with
$J_{0}=\partial_{x}^{3}+4u_{0}\partial_{x}+2u_{0x},\;\;J_{1}=4u_{1}\partial_{x}+2\,u_{1x},\;\;J_{2}=4u_{2}\partial_{x}+2\,u_{2x},\;\;J_{3}=-4\partial_{x}$.

For $P=P^{(n)}$, we have

$u_{0t_{n}}+\lambda
u_{1t_{n}}+\lambda^{2}u_{2t_{n}}=J_{0}P_{n}+\lambda(J_{0}P_{n-1}+J_{1}P_{n})+\lambda^{2}(J_{0}P_{n-2}+J_{1}P_{n-1}+J_{2}P_{n}),$
(3c)

which can be written in vector form

$\displaystyle\left(\begin{array}[]{c}u_{0}\\\ u_{1}\\\
u_{2}\end{array}\right)_{t_{n}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&J_{0}\\\ 0&J_{0}&J_{1}\\\
J_{0}&J_{1}&J_{2}\end{array}\right)\left(\begin{array}[]{l}P_{n-2}\\\
P_{n-1}\\\ P_{n}\end{array}\right)=\left(\begin{array}[]{ccc}0&J_{0}&0\\\
J_{0}&J_{1}&0\\\
0&0&-J_{3}\end{array}\right)\left(\begin{array}[]{l}P_{n-1}\\\ P_{n}\\\
P_{n+1}\end{array}\right)$ (3s) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}J_{0}&0&0\\\ 0&-J_{2}&-J_{3}\\\
0&-J_{3}&0\end{array}\right)\left(\begin{array}[]{l}P_{n}\\\ P_{n+1}\\\
P_{n+2}\end{array}\right)=\left(\begin{array}[]{ccc}-J_{1}&-J_{2}&-J_{3}\\\
-J_{2}&-J_{3}&0\\\
-J_{3}&0&0\end{array}\right)\left(\begin{array}[]{l}P_{n+1}\\\ P_{n+2}\\\
P_{n+3}\end{array}\right),$ (3ag)

with the first being given directly by (3c). The general recursion (3b) then
gives the remaining representations.

These matrices define the 4 compatible Hamiltonian operators of this cKdV
hierarchy, labelled respectively as $B_{3},\ B_{2},\,B_{1}$ and $B_{0}$.
Defining

$\left(\begin{array}[]{c}P_{n-2}\\\ P_{n-1}\\\ P_{n}\\\
\end{array}\right)=\delta_{u}H_{n}=\left(\begin{array}[]{c}\delta_{u_{0}}H_{n}\\\
\delta_{u_{1}}H_{n}\\\ \delta_{u_{2}}H_{n}\end{array}\right),$ (4a) we have
${\bf
u}_{t_{n}}=B_{3}\delta_{u}H_{n}=B_{2}\delta_{u}H_{n+1}=B_{1}\delta_{u}H_{n+2}=B_{0}\delta_{u}H_{n+3},\quad\mbox{for}\;\;\;n\geq
0,$ (4b) where ${\bf u}=(u_{0},u_{1},u_{2})^{T}$ and $\displaystyle
H_{0}=u_{2},\;\;H_{1}=\frac{1}{4}(4u_{1}+u_{2}^{2}),\;\;H_{2}=u_{0}+\frac{1}{8}u_{2}(4u_{1}+u_{2}^{2}),\;\;H_{3}=\frac{1}{2}u_{0}u_{2}+\frac{1}{64}(4u_{1}+u_{2}^{2})(4u_{1}+5u_{2}^{2}),$
$\displaystyle
H_{4}=\frac{1}{8}u_{0}(4u_{1}+3u_{2}^{2})+\frac{1}{128}u_{2}(4u_{1}+u_{2}^{2})(12u_{1}+7u_{2}^{2})-\frac{1}{16}u_{2x}^{2},$
(4c) $\displaystyle
H_{5}=\frac{1}{16}u_{0}(4u_{0}+12u_{1}u_{2}+5u_{2}^{3})+\frac{1}{512}(4u_{1}+u_{2}^{2})(16u_{1}^{2}+56u_{1}u_{2}^{2}+21u_{2}^{4})-\frac{1}{32}u_{2x}(4u_{1x}+5u_{2}u_{2x}).$
We also have
$B_{0}\delta_{u}H_{0}=B_{0}\delta_{u}H_{1}=B_{0}\delta_{u}H_{2}=B_{1}\delta_{u}H_{0}=B_{1}\delta_{u}H_{1}=B_{2}\delta_{u}H_{0}=0,$
(4d) meaning that $B_{0}$ has three local Casimir functions
$H_{0},\,H_{1},\,H_{2}$, whilst $B_{1}$ has two local Casimir functions
$H_{0},\ H_{1}$, with $B_{2}$ having just one local Casimir function $H_{0}$.

###### Remark 2.1

Corresponding to these densities, we have an array of fluxes $F_{nm}$, in
local conservation laws, given by the $t_{n}$ evolution of $H_{m}$:

$\partial_{t_{n}}H_{m}=\partial_{x}F_{nm}.$

In [12] we used these to build first integrals for stationary flows, but the
same integrals arise from the Lax representation as well as the multi-
Hamiltonian ladders, so we omit the formulae for $F_{nm}$ here.

####  2.1.1 Miura Maps

In [2], Miura maps were presented for the entire class of systems described by
the “energy dependent” Schrödinger operator. In the $M=3$ case, there are 4
sets of variables ${\bf u},\;{\bf w}=(w_{0},w_{1},w_{2})^{T},\;{\bf
v}=(v_{0},v_{1},v_{2})^{T}$ and ${\bf z}=(z_{0},z_{1},z_{2})^{T}$, related by
the following Miura maps

$\begin{array}[]{llll}M_{1}[{\bf
w}]:&u_{0}=-w_{0x}-w_{0}^{2},&u_{1}=w_{1},&u_{2}=w_{2},\\\ M_{2}[{\bf
v}]:&w_{0}=v_{0},&w_{1}=-v_{1x}-2v_{0}v_{1},&w_{2}=v_{2},\\\ M_{3}[{\bf
z}]:&v_{0}=z_{0},&v_{1}=z_{1},&v_{2}=-z_{2x}-2z_{0}z_{2}-z_{1}^{2}.\end{array}$
(5a) We have the 4 local Hamiltonian operators $B_{i}^{u}\equiv B_{i}$, given
by (4b), with 3 local operators $B_{3}^{w}$, $B_{2}^{w}$, $B_{1}^{w}$ in the
$w-$space, 2 local operator $B_{3}^{v}$, $B_{2}^{v}$ in the $v-$space and just
1 local operator $B_{3}^{z}$ in the $z-$space. These are depicted in Figure 1
and related by $\displaystyle
B_{k}^{u}=\frac{Du}{Dw}B_{k}^{w}\left(\frac{Du}{Dw}\right)^{\dagger}\;\;\;\mbox{for}\;\;\;k=1,2,3,$
(5b) $\displaystyle
B_{k}^{w}=\frac{Dw}{Dv}B_{k}^{v}\left(\frac{Dw}{Dv}\right)^{\dagger}\;\;\;\mbox{for}\;\;\;k=2,3\qquad\mbox{and}\qquad
B_{3}^{v}=\frac{Dv}{Dz}B_{3}^{z}\left(\frac{Dv}{Dz}\right)^{\dagger},$ where
$\frac{Du}{Dw}$, $\frac{Dw}{Dv}$ and $\frac{Dv}{Dz}$ are the Jacobians of the
maps $M_{1}[{\bf w}]$, $M_{2}[{\bf v}]$ and $M_{3}[{\bf z}]$, respectively,
given in (5a), and the Hamiltonian operators in $w-$, $v-$ and $z-$space take
the form $\displaystyle
B_{3}^{w}=\left(\begin{array}[]{ccc}0&0&-\partial_{x}^{2}+2\partial_{x}w_{0}\\\
0&\partial_{x}^{3}-4(w_{0}^{2}+w_{0x})\partial_{x}-2(2w_{0}w_{0x}+w_{0xx})&4w_{1}\partial_{x}+2w_{1x}\\\
\partial_{x}^{2}+2w_{0}\partial_{x}&4w_{1}\partial_{x}+2w_{1x}&4w_{2}\partial_{x}+2w_{2x}\end{array}\right),$
(5f) $\displaystyle
B_{2}^{w}=\left(\begin{array}[]{ccc}0&-\partial_{x}^{2}+2\partial_{x}w_{0}&0\\\
\partial_{x}^{2}+2w_{0}\partial_{x}&4w_{1}\partial_{x}+2w_{1x}&0\\\
0&0&4\partial_{x}\end{array}\right),\quad
B_{1}^{w}=\left(\begin{array}[]{ccc}-\partial_{x}&0&0\\\
0&-4w_{2}\partial_{x}-2w_{2x}&4\partial_{x}\\\
0&4\partial_{x}&0\end{array}\right),$ (5m) $\displaystyle
B_{3}^{v}=\left(\begin{array}[]{ccc}0&0&-\partial_{x}^{2}\\!+\\!2\partial_{x}v_{0}\\\
0&-\partial_{x}&2\partial_{x}v_{1}\\\
\partial_{x}^{2}\\!+\\!2v_{0}\partial_{x}&2v_{1}\partial_{x}&4v_{2}\partial_{x}\\!+\\!2v_{2x}\end{array}\right),\;\;B_{2}^{v}=\left(\begin{array}[]{ccc}0&-\partial_{x}&0\\\
-\partial_{x}&0&0\\\
0&0&4\partial_{x}\end{array}\right),\;\;B_{3}^{z}=\left(\begin{array}[]{ccc}0&0&-\partial_{x}\\\
0&-\partial_{x}&0\\\ -\partial_{x}&0&0\end{array}\right).$ (5w)

Figure 1: Hamiltonian Operators in the 4
Spaces$B_{3}^{u}$$B_{2}^{u}$$B_{1}^{u}$$B_{0}^{u}$$B_{3}^{w}$$B_{2}^{w}$$B_{1}^{w}$$B_{3}^{v}$$B_{2}^{v}$$B_{3}^{z}$

We have a sequence of Hamiltonians $H_{k}^{u}\equiv H_{k}$, given by (4), as
functions of $u_{i}$ and their $x-$derivatives. The Miura maps then define
$H_{k}^{w}$, $H_{k}^{v}$ and $H_{k}^{z}$, by substituting the formulae (5a)
into $H_{k}^{u}$. The flow ${\bf u}_{t_{n}}$, defined by (4b), gives rise to
the flows:

$\displaystyle{\bf
w}_{t_{n}}=B_{3}^{w}\delta_{w}H_{n}^{w}=B_{2}^{w}\delta_{w}H_{n+1}^{w}=B_{1}^{w}\delta_{w}H_{n+2}^{w},$
(6) $\displaystyle{\bf
v}_{t_{n}}=B_{3}^{v}\delta_{v}H_{n}^{v}=B_{2}^{v}\delta_{v}H_{n+1}^{v}\qquad\mbox{and}\qquad{\bf
z}_{t_{n}}=B_{3}^{z}\delta_{z}H_{n}^{z}.$

###  2.2 The Stationary Flows

A stationary flow of (4b) (for $t_{n}$) means that we reduce to a finite
dimensional space with ${\bf u}_{t_{n}}=0$. The “time” variable for this
system is $x$, which is the variable which appears in the Lagrangians, given
below.

  1. 1.

Using $B_{0}$ and in view of
$B_{0}\delta_{u}H_{0}=B_{0}\delta_{u}H_{1}=B_{0}\delta_{u}H_{2}=0$, we have

$B_{0}\delta_{u}H_{n+3}=0\quad\Rightarrow\quad\delta_{u}(H_{n+3}-\alpha_{0}H_{0}-4\alpha_{1}H_{1}-8\alpha_{2}H_{2})=0,$
(7a) which gives the first Lagrangian: ${\cal
L}_{n+3}=H_{n+3}-\alpha_{0}u_{2}-\alpha_{1}(4u_{1}+u_{2}^{2})-\alpha_{2}(8u_{0}+4u_{1}u_{2}+u_{2}^{3}).$
(7b)

We then use the (generalised) Legendre transformation to find canonical
coordinates $(q_{i},p_{i})$ and the Hamiltonian function.

  2. 2.

Using $B_{1}$ and the squared eigenfunction representation (following [4]), we
have

$\delta_{u_{0}}H_{n+2}=-\frac{1}{2}\varphi^{2}\quad\Rightarrow\quad\varphi^{2}(J_{0}\varphi^{2})=\left(2\varphi^{3}(\varphi_{xx}+u_{0}\varphi)\right)_{x}=0\quad\Rightarrow\quad
2\varphi^{3}(\varphi_{xx}+u_{0}\varphi)=4\beta,$ (8a) and
$\delta_{u_{1}}H_{n+2}=4b_{1},\quad\delta_{u_{2}}H_{n+2}=2b_{1}u_{2}+b_{0}.$
(8b) The exact derivative of (8a) arises as a consequence of the skew symmetry
of $J_{0}$. We can write the above equations as variational derivatives of a
single Lagrangian function: ${\cal
L}_{n+2}=H_{n+2}-b_{0}u_{2}-b_{1}(4u_{1}+u_{2}^{2})+\frac{1}{2}u_{0}\varphi^{2}-\frac{1}{2}\,\varphi_{x}^{2}+\frac{\beta}{\varphi^{2}},$
(8c) with the above equations being given by $\delta_{u}{\cal
L}_{n+2}=0,\;\;\delta_{\varphi}{\cal L}_{n+2}=0$.

Again, the (generalised) Legendre transformation gives canonical coordinates
$(Q_{i},P_{i})$ and the Hamiltonian function.

###### Remark 2.2

The Hamiltonians listed in (4) are built algorithmically, so often have
inconvenient coefficients. For our stationary flows, we typically choose a
constant multiple of these in the formulae (7b) and (8c).

Since $J_{0}$ is a linear operator we can extend the squared eigenfunction
representation (8a) to include multiple eigenfunctions (see [12] for more
details), with

$\delta_{u_{0}}H_{n+2}=-\frac{1}{2}\sum_{i}\varphi_{i}^{2},$ (9a) which yields
$\delta_{u}{\cal L}_{n+2}=0,\;\;\;\delta_{\varphi_{i}}{\cal L}_{n+2}=0$, with
${\cal
L}_{n+2}=H_{n+2}-b_{0}u_{2}-b_{1}(4u_{1}+u_{2}^{2})+\sum_{i}\left(\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{1}{2}\,\varphi_{ix}^{2}+\frac{\beta_{i}}{\varphi_{i}^{2}}\right),$
(9b) or ${\cal
L}_{n+2}=H_{n+2}-b_{0}u_{2}-b_{1}(4u_{1}+u_{2}^{2})+\sum_{i}\left(\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{1}{2}\,\varphi_{ix}^{2}\right)+\frac{\beta}{{\bf\varphi}^{2}},$
(9c) where $\varphi^{2}=\sum_{i}\varphi_{i}^{2}$.

###### Remark 2.3

The Lagrangian in (9c) is rotationally invariant in the $\varphi$ space, so
the angular momenta $(\varphi_{i}\varphi_{jx}-\varphi_{j}\varphi_{ix})$ are
constants of the motion. These can be generalised for the case of (9b).

####  2.2.1 The Lax Representation of a Stationary Flow

Setting $U_{t_{n}}=0$ in (2d), we obtain

$V^{(n)}_{x}=\left[U,V^{(n)}\right],$ (10)

with these matrices being written in terms of $u_{i}$ and their
$x-$derivatives. The above Legendre transformations give us formulae for these
variables in terms of the canonical coordinates, in terms of which (10) gives
a Lax representation for the stationary flow, with $V^{(n)}$ playing the role
of the Lax matrix $L$.

##  3 The Stationary $t_{1}$ Flows

We first build the Lagrangians and Hamiltonians. We use the corresponding Lax
pair to construct the first integrals. A bi-Hamiltonian representation is then
constructed.

###  3.1 Lagrangians and Legendre Transformations

We build 3 different Hamiltonians.

##### The Lagrangian (7b),

using $8H_{4}$ and labelling $-{\cal L}_{4}$ as $\cal L$, gives

${\cal
L}=\frac{1}{2}u_{2x}^{2}-4u_{0}u_{1}-3u_{2}\left(u_{0}u_{2}+u_{1}^{2}\right)-\frac{1}{16}u_{2}^{3}\left(40u_{1}+7u_{2}^{2}\right)+\alpha_{0}u_{2}+\alpha_{1}(4u_{1}+u_{2}^{2})+\alpha_{2}(8u_{0}+4u_{1}u_{2}+u_{2}^{3}).$
(11a) This Lagrangian is degenerate, but $\delta_{u_{0}}{\cal
L}=\delta_{u_{1}}{\cal L}=0\quad\Rightarrow\quad
u_{0}=\alpha_{1}-2\alpha_{2}u_{2}+\frac{1}{2}u_{2}^{3},\;\;\;u_{1}=2\alpha_{2}-\frac{3}{4}u_{2}^{2},$
after which ${\cal
L}=\frac{1}{2}u_{2x}^{2}+(\alpha_{0}-4\alpha_{2}^{2})u_{2}-2\alpha_{1}u_{2}^{2}+2\alpha_{2}u_{2}^{3}-\frac{1}{4}u_{2}^{5}+8\alpha_{1}\alpha_{2}$,
which gives the Hamiltonian
$h^{(q)}=\frac{1}{2}p_{1}^{2}+(4\alpha_{2}^{2}-\alpha_{0})q_{1}+2\alpha_{1}q_{1}^{2}-2\alpha_{2}q_{1}^{3}+\frac{1}{4}q_{1}^{5},$
(11b) where $q_{1}=u_{2},\,p_{1}=u_{2x}$.

##### The Lagrangian (9b),

using $H_{3}$ and labelling $-{\cal L}_{3}$ as $\tilde{\cal L}$, gives

$\tilde{\cal
L}=\sum_{i=1}^{N}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{\beta_{i}}{\varphi_{i}^{2}}\right)+b_{0}u_{2}+b_{1}(4u_{1}+u_{2}^{2})-\frac{1}{2}u_{0}u_{2}-\frac{1}{64}(4u_{1}+u_{2}^{2})(4u_{1}+5u_{2}^{2}).$
(12a) This is also degenerate, but $\delta_{u_{i}}\tilde{\cal L}=0$, with
$i=0,1,2$, give
$u_{0}=8b_{1}\varphi^{2}-\frac{1}{2}\varphi^{6}+2b_{0},\;\;u_{1}=-\frac{3}{4}\varphi^{4}+8b_{1},\;\;u_{2}=-\varphi^{2},\qquad\mbox{where}\quad\varphi^{2n}=\left(\sum_{i=1}^{N}\varphi_{i}^{2}\right)^{n},$
leading to $\tilde{\cal
L}=\sum_{i=1}^{N}\left(\frac{1}{2}\varphi_{ix}^{2}-b_{0}\varphi_{i}^{2}-\frac{\beta_{i}}{\varphi_{i}^{2}}\right)-2b_{1}\varphi^{4}+\frac{1}{16}\varphi^{8}+16b_{1}^{2}.$
(12b) Defining $Q_{i}=\varphi_{i},\,P_{i}=\varphi_{ix}$ and removing the
additive constant, we obtain the Hamiltonian
$h^{(Q)}=\sum_{i=1}^{N}\left(\frac{1}{2}P_{i}^{2}+b_{0}Q_{i}^{2}+\frac{\beta_{i}}{Q_{i}^{2}}\right)+2b_{1}{\bf
Q}^{4}-\frac{1}{16}{\bf Q}^{8},\qquad\mbox{where}\quad{\bf
Q}^{2n}=\left(\sum_{i=1}^{N}Q_{i}^{2}\right)^{n}.$ (12c)

##### The Lagrangian (9c)

gives (removing an additive constant)

$h^{(Q)}=\sum_{i=1}^{N}\left(\frac{1}{2}P_{i}^{2}+b_{0}Q_{i}^{2}\right)+2b_{1}{\bf
Q}^{4}-\frac{1}{16}{\bf Q}^{8}+\frac{\beta}{{\bf Q}^{2}}.$ (13a)

Clearly, this Hamiltonian is rotationally invariant, so the angular momenta
are first integrals.

###  3.2 The Lax Representation

We now use the stationary Lax representation, given by (10), with the specific
form $P^{(1)}=\lambda+\frac{1}{2}u_{2}$, defined by (2c), with (3a).
Specifically, for $M=3$, we have

$\displaystyle U=\left(\begin{array}[]{cc}0&1\\\
\lambda^{3}-u_{2}\lambda^{2}-u_{1}\lambda-u_{0}&0\end{array}\right),\qquad
V^{(1)}=\frac{1}{2}\left(\begin{array}[]{cc}-u_{2x}&4\lambda+2u_{2}\\\
a_{21}&u_{2x}\end{array}\right),$ (14e) where $\displaystyle
a_{21}=2(\lambda^{3}-u_{2}\lambda^{2}-u_{1}\lambda-
u_{0})(2\lambda+u_{2})-u_{2xx}.$

##### Using the coordinates of the Hamiltonian (11b),

we define $L^{(1)}=-2V^{(1)}$, with

$\left.\begin{array}[]{l}u_{0}=\alpha_{1}-2\alpha_{2}q_{1}+\frac{1}{2}q_{1}^{3},\\\
u_{1}=2\alpha_{2}-\frac{3}{4}q_{1}^{2},\;u_{2}=q_{1},\\\
u_{2xx}=q_{1xx}=-\frac{\partial h^{(q)}}{\partial
q_{1}}\end{array}\right\\}\quad\Rightarrow\quad
L^{(1)}=\left(\begin{array}[]{cc}p_{1}&-4\lambda-2q_{1}\\\
a_{21}&-p_{1}\end{array}\right),$ (15a) where $a_{21}$ is given by (LABEL:Lax-
UV1) after substituting canonical coordinates.

The characteristic equation of $L^{(1)}$ is

$z^{2}-16\lambda^{5}+32\alpha_{2}\lambda^{3}+16\alpha_{1}\lambda^{2}+4(\alpha_{0}-4\alpha_{2}^{2})\lambda-2h^{(q)}=0,$
(15b)

with $h^{(q)}$ given by (11b).

The Lax representation of the equation generated by $h^{(q)}$ is

$L^{(1)}_{x}=\\{L^{(1)},h^{(q)}\\}=[U,L^{(1)}],$ (15c)

with $U$ given by (LABEL:Lax-UV1), after substituting for $u_{i}$.

##### Using the coordinates of the Hamiltonian (12c),

we define $L^{(1)}=V^{(1)}$ to obtain

$\left.\begin{array}[]{l}u_{0}=8b_{1}{\bf Q}^{2}-\frac{1}{2}{\bf
Q}^{6}+2b_{0},\\\ u_{1}=8b_{1}-\frac{3}{4}{\bf Q}^{4},\;u_{2}=-{\bf Q}^{2}\\\
u_{2xx}=-2\sum_{i=1}^{N}\left(P_{i}^{2}-Q_{i}\frac{\partial h^{(Q)}}{\partial
Q_{i}}\right)\end{array}\right\\}\quad\Rightarrow\quad
L^{(1)}=\left(\begin{array}[]{cc}\sum_{1}^{N}Q_{i}P_{i}&2\lambda-{\bf
Q}^{2}\\\ a_{21}&-\sum_{1}^{N}Q_{i}P_{i}\end{array}\right),$ (16a) with
$a_{21}=\frac{1}{4}\lambda(2\lambda+{\bf Q}^{2})(4\lambda^{2}+{\bf
Q}^{4}-32b_{1})-4b_{0}\lambda+\sum_{1}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)$.

The characteristic equation of $L^{(1)}$ is

$z^{2}-4\lambda^{5}+32b_{1}\lambda^{3}+8b_{0}\lambda^{2}-4h^{(Q)}\lambda+2\Phi^{(Q)}=0,$
(16b)

where $h^{(Q)}$ is given by (12c) and

$\Phi^{(Q)}=\frac{1}{2}\sum_{1\leq i<j\leq
N}h_{ij}+\sum_{i=1}^{N}\beta_{i},\quad\mbox{where}\quad
h_{ij}=(Q_{i}P_{j}-Q_{j}P_{i})^{2}+2\left(\beta_{i}\frac{Q_{j}^{2}}{Q_{i}^{2}}+\beta_{j}\frac{Q_{i}^{2}}{Q_{j}^{2}}\right).$
(16c)

Indeed, each $h_{ij}$ is a first integral and, being deformations of the
(squares of) angular momenta, it is straightforward to build an involutive set
of integrals, so (12c) defines a superintegrable system (for $N\geq 3$).

This is a multi-component version of Case 1, Table II of [8], but with a
particular form of potential. However, this particular case has a Lax
representation.

When $N=1$, $\Phi^{(Q)}=\beta_{1}$. When $N=2$, we have 2 integrals
$h^{(Q)},\,h_{12}$.

When $N=3$, we have 4 integrals $h^{(Q)},\,h_{ij}$, with $h^{(Q)},\,h_{12}$
and $\Phi^{(Q)}$ in involution.

The Lax equations generated by $h^{(Q)}$ and $\Phi^{(Q)}$ are

$L^{(1)}_{x}=\\{L^{(1)},h^{(Q)}\\}=[U,L^{(1)}],\quad\\{L^{(1)},\Phi^{(Q)}\\}=0,\quad\mbox{with}\quad
U=\left(\begin{array}[]{cc}0&1\\\ b_{21}&0\end{array}\right),$ (16d)

where $b_{21}=\frac{1}{2}(\lambda+{\bf Q}^{2})(2\lambda^{2}+{\bf
Q}^{4}-16b_{1})+\frac{1}{4}\lambda{\bf Q}^{4}-2b_{0}$.

In fact, $\\{L^{(1)},h_{ij}\\}=0$, for each of the functions $h_{ij}$ of
(16c).

###### Remark 3.1

It is known (see [5]) that the general Hamiltonian (1a) is superintegrable,
but not maximal, having $2N-3$ additional independent integrals. However, the
specific potentials derived as stationary reductions of the cKdV hierarchy
also have a Lax representation.

##### Using the coordinates of (13a),

we have the Lax representation (16a) with
$\sum_{1}^{N}\frac{\beta_{i}}{Q_{i}^{2}}$ replaced by $\frac{\beta}{{\bf
Q}^{2}}$ and the integral $\Phi^{(Q)}$ reduces to
$\frac{1}{2}\sum_{i<j}(Q_{i}P_{j}-Q_{j}P_{i})^{2}+\beta$, reflecting the
rotational symmetry.

###  3.3 Bi-Hamiltonian Formulation when $N=1$

Here we have

$h^{(Q)}=\frac{1}{2}P_{1}^{2}+b_{0}Q_{1}^{2}+2b_{1}Q_{1}^{4}+\frac{\beta_{1}}{Q_{1}^{2}}-\frac{1}{16}Q_{1}^{8}.$
(17)

The Hamiltonians (11b) and (17) define coordinates ${\bf
q}=(q_{1},p_{1},\alpha_{0},\alpha_{1},\alpha_{2})$ and ${\bf
Q}=(Q_{1},P_{1},\beta_{1},b_{0},b_{1})$, respectively. From the definitions of
the canonical coordinates, we have the following Poisson map

$q_{1}=-Q_{1}^{2},\quad
p_{1}=-2Q_{1}P_{1},\quad\alpha_{0}=-4h^{(Q)}+64b_{1}^{2},\quad\alpha_{1}=2b_{0},\quad\alpha_{2}=4b_{1},$
(18)

with $h^{(q)}=-4\beta_{1}$.

###### Remark 3.2

The formulae for $\alpha_{i}$ and $h^{(q)}$ (in terms of the variables ${\bf
Q}$) can be read directly from the formulae for the coefficients of the
characteristic equations of the Lax matrices in the respective coordinates.

We then follow the standard approach (outlined in Section 3.1.1 of [12]) to
build a bi-Hamiltonian representation.

In the 5D spaces with coordinates $\bf q$ and $\bf Q$, we introduce the
degenerate extension of the canonical Poisson tensor:

${\cal P}_{0}^{(q)}={\cal
P}_{1}^{(Q)}=\left(\begin{array}[]{rrrrr}0&1&0&0&0\\\ -1&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),\quad\mbox{with}\;\;\;\frac{d{\bf
q}}{dt_{f}}={\cal P}_{0}^{(q)}\nabla_{q}f\ \mbox{and}\ \frac{d{\bf
Q}}{dt_{F}}={\cal P}_{1}^{(Q)}\nabla_{Q}F,$ (19a) for any function $f(\bf{q})$
and $F(\bf{Q})$. The formulae (18) represent a mapping from the $\bf Q$ space
to the $\bf q$ space. We use the Jacobian of the inverse map to construct a
Poisson tensor ${\cal P}_{0}^{(Q)}=-4\frac{\partial{\bf Q}}{\partial{\bf
q}}\,{\cal P}_{0}^{(q)}\,\left(\frac{\partial{\bf Q}}{\partial{\bf
q}}\right)^{T}$ in the $\bf Q$ space, whose explicit form is given by ${\cal
P}_{0}^{(Q)}=\left(\begin{array}[]{ccccc}0&-\frac{1}{Q_{1}^{2}}&a_{13}&0&0\\\
\frac{1}{Q_{1}^{2}}&0&a_{23}&0&0\\\ -a_{13}&-a_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),$ (19b) where $(a_{13},a_{23},0,0,0)^{T}={\cal
P}_{1}^{(Q)}\nabla_{Q}h^{(Q)}$. This is compatible with the canonical bracket
${\cal P}_{1}^{(Q)}$ on the ${\bf Q}$ space, which in turn allows us to
construct a Poisson tensor ${\cal
P}_{1}^{(q)}=-\frac{1}{4}\,\frac{\partial{\bf q}}{\partial{\bf Q}}\,{\cal
P}_{1}^{(Q)}\left(\frac{\partial{\bf q}}{\partial{\bf Q}}\right)^{T}$, where
${\cal P}_{1}^{(q)}=\left(\begin{array}[]{ccccc}0&q_{1}&b_{13}&0&0\\\
-q_{1}&0&b_{23}&0&0\\\ -b_{13}&-b_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),$ (19c)

with $(b_{13},b_{23},0,0,0)^{T}={\cal P}_{0}^{(q)}\nabla_{q}h^{(q)}$.

Each of these Poisson matrices has three Casimirs and the $t_{1}$ ($=t_{h}$)
flow in each space has a bi-Hamiltonian representation:

$\displaystyle{\bf q}_{t_{h}}={\cal P}_{0}^{(q)}\nabla_{q}h^{(q)}={\cal
P}_{1}^{(q)}\nabla_{q}\,\alpha_{0},$ (20a) $\displaystyle{\cal
P}_{0}^{(q)}\nabla_{q}\alpha_{0}={\cal P}_{0}^{(q)}\nabla_{q}\alpha_{1}={\cal
P}_{0}^{(q)}\nabla_{q}\alpha_{2}={\cal P}_{1}^{(q)}\nabla_{q}h^{(q)}={\cal
P}_{1}^{(q)}\nabla_{q}\alpha_{1}={\cal P}_{1}^{(q)}\nabla_{q}\alpha_{2}=0,$ in
the ${\bf q}$ space and $\displaystyle{\bf Q}_{t_{h}}={\cal
P}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\cal P}_{0}^{(Q)}\nabla_{Q}\beta_{1},$ (20b)
$\displaystyle{\cal P}_{1}^{(Q)}\nabla_{Q}b_{0}={\cal
P}_{1}^{(Q)}\nabla_{Q}b_{1}={\cal P}_{1}^{(Q)}\nabla_{Q}\beta_{1}={\cal
P}_{0}^{(Q)}\nabla_{Q}b_{0}={\cal P}_{0}^{(Q)}\nabla_{Q}b_{1}={\cal
P}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0,$ in the ${\bf Q}$ space.

##  4 The Stationary $t_{2}$ Flows

We first build the Lagrangians and Hamiltonians. We use the corresponding Lax
pair to construct the first integrals. A bi-Hamiltonian representation is then
constructed.

###  4.1 Lagrangians and Legendre Transformations

We build 3 different Hamiltonians.

##### The Lagrangian (7b),

using $2H_{5}$ and labelling ${\cal L}_{5}$ as $\cal L$, gives

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{2}u_{0}^{2}+\frac{1}{8}u_{0}u_{2}\left(12u_{1}+5u_{2}^{2}\right)+\frac{1}{256}\left(4u_{1}+u_{2}^{2}\right)\left(16u_{1}^{2}+56u_{1}u_{2}^{2}+21u_{2}^{4}\right)-\frac{1}{4}u_{1x}u_{2x}$
(21a) $\displaystyle\hskip
113.81102pt-\frac{5}{16}u_{2}u_{2x}^{2}-\alpha_{0}u_{2}-\alpha_{1}(4u_{1}+u_{2}^{2})-\alpha_{2}(8u_{0}+4u_{1}u_{2}+u_{2}^{3}).$
This Lagrangian is degenerate, but $\delta_{u_{0}}{\cal L}=0$ gives
$u_{0}=8\alpha_{2}-\frac{3}{2}u_{1}u_{2}-\frac{5}{8}u_{2}^{3}$, after which
$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\frac{1}{4}u_{1}^{3}-\frac{3}{16}u_{1}^{2}u_{2}^{2}-\frac{25}{64}u_{1}u_{2}^{4}-\frac{29}{256}u_{2}^{6}-\frac{1}{4}u_{1x}u_{2x}-\frac{5}{16}u_{2}u_{2x}^{2}$
(21b) $\displaystyle\hskip
85.35826pt-\alpha_{0}u_{2}-\alpha_{1}(4u_{1}+u_{2}^{2})+4\alpha_{2}u_{2}\left(2u_{1}+u_{2}^{2}\right)-32\alpha_{2}^{2},$
leading to
$q_{1}=u_{1},\;q_{2}=u_{2},\;p_{1}=-\frac{1}{4}u_{2x},\;p_{2}=-\frac{5}{8}u_{2}u_{2x}-\frac{1}{4}u_{1x}$
and (removing the additive constant) the Hamiltonian
$h^{(q)}=5q_{2}p_{1}^{2}-4p_{1}p_{2}+\alpha_{0}q_{2}+\alpha_{1}(4q_{1}+q_{2}^{2})-4\alpha_{2}q_{2}(2q_{1}+q_{2}^{2})-\frac{1}{4}q_{1}^{3}+\frac{3}{16}q_{1}^{2}q_{2}^{2}+\frac{25}{64}q_{1}q_{2}^{4}+\frac{29}{256}q_{2}^{6}.$
(21c)

##### The Lagrangian (9b),

using $8H_{4}$ and labelling $-{\cal L}_{4}$ as $\tilde{\cal L}$, we find

$\tilde{\cal
L}=\sum_{i=1}^{N-1}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{2}u_{0}\varphi_{i}^{2}-\frac{\beta_{i+1}}{\varphi_{i}^{2}}\right)+\frac{1}{2}u_{2x}^{2}-\frac{u_{2}}{16}\left(4u_{1}+u_{2}^{2}\right)\left(12u_{1}+7u_{2}^{2}\right)-u_{0}\left(4u_{1}+3u_{2}^{2}\right)+b_{0}u_{2}+b_{1}(4u_{1}+u_{2}^{2}).$
(22a) This is also degenerate, but $\delta_{u_{i}}\tilde{\cal L}=0$ with
$i=0,1$ give
$u_{0}=\frac{3}{16}u_{2}\sum_{1}^{N-1}\varphi_{i}^{2}+\frac{1}{2}u_{2}^{3}+b_{1},\,u_{1}=-\frac{1}{8}\sum_{1}^{N-1}\varphi_{i}^{2}-\frac{3}{4}u_{2}^{2}$,
leading to $\tilde{\cal
L}=\sum_{i=1}^{N-1}\left(\frac{1}{2}\varphi_{ix}^{2}-\frac{1}{4}(u_{2}^{3}+2b_{1})\varphi_{i}^{2}-\frac{\beta_{i+1}}{\varphi_{i}^{2}}\right)+\frac{1}{2}u_{2x}^{2}-\frac{3}{64}u_{2}\left(\sum_{i=1}^{N-1}\varphi_{i}^{2}\right)^{2}-\frac{1}{4}u_{2}(u_{2}^{4}+8b_{1}u_{2}-4b_{0}).$
(22b) Defining
$Q_{1}=u_{2},\,Q_{i+1}=\varphi_{i},\,P_{1}=u_{2x},\,P_{i+1}=\varphi_{ix}$, for
$i=1,\dots,N-1$, leads to the Hamiltonian
$h^{(Q)}=\frac{1}{2}P_{1}^{2}+\frac{1}{2}\sum_{i=2}^{N}\left(P_{i}^{2}+\frac{2\beta_{i}}{Q_{i}^{2}}\right)-b_{0}Q_{1}+\frac{1}{2}b_{1}(4Q_{1}^{2}+{\bf
Q}^{2})+\frac{1}{64}Q_{1}(4Q_{1}^{2}+{\bf Q}^{2})(4Q_{1}^{2}+3{\bf Q}^{2}).$
(22c)

where ${\bf Q}^{2}=\sum_{i=2}^{N}Q_{i}^{2}$.

##### The Lagrangian (9c)

gives

$h^{(Q)}=\frac{1}{2}\sum_{i=1}^{N}P_{i}^{2}-b_{0}Q_{1}+\frac{1}{2}b_{1}(4Q_{1}^{2}+{\bf
Q}^{2})+\frac{1}{64}Q_{1}(4Q_{1}^{2}+{\bf Q}^{2})(4Q_{1}^{2}+3{\bf
Q}^{2})+\frac{\beta}{{\bf Q}^{2}}.$ (23)

Clearly, this Hamiltonian is rotationally invariant in the $Q_{2},\dots,Q_{N}$
space, so the angular momenta in this space are first integrals.

###  4.2 The Lax Representation

Again, we use the stationary Lax representation, given by (10), with the
specific form
$P^{(2)}=\lambda^{2}+\frac{1}{2}u_{2}\lambda+\frac{1}{8}(4u_{1}+3u_{2}^{2})$,
defined by (2c) and (3a), giving

$V^{(2)}=\frac{1}{4}\left(\begin{array}[]{cc}-(2\lambda+3u_{2})u_{2x}-2u_{1x}&8\lambda^{2}+4u_{2}\lambda+3u_{2}^{2}+4u_{1}\\\
b_{21}&(2\lambda+3u_{2})u_{2x}+2u_{1x}\end{array}\right),$ (24)

with $b_{21}=(\lambda^{3}-u_{2}\lambda^{2}-u_{1}\lambda-
u_{0})(8\lambda^{2}+4u_{2}\lambda+3u_{2}^{2}+4u_{1})-2\lambda
u_{2xx}-(3u_{2}u_{2x}+2u_{1x})_{x}$.

We also need $U$ and $V^{(1)}$ of (LABEL:Lax-UV1), but now written in terms of
the stationary $t_{2}$ coordinates.

##### Using the coordinates of the Hamiltonian (21c),

we define $L^{(2)}=\frac{1}{2}V^{(2)}$ to obtain

$\left.\begin{array}[]{l}u_{0}=8\alpha_{2}-\frac{3}{2}q_{1}q_{2}-\frac{5}{8}q_{2}^{3},\\\
u_{1}=q_{1},\;u_{2}=q_{2},\;u_{2x}=-4p_{1}\\\
u_{1x}=2(5q_{2}p_{1}-2p_{2})\end{array}\right\\}\quad\Rightarrow\quad
L^{(2)}=\left(\begin{array}[]{cc}\lambda
p_{1}+p_{2}-q_{2}p_{1}&\lambda^{2}+\frac{1}{2}q_{2}\lambda+\frac{1}{8}(4q_{1}+3q_{2}^{2})\\\
a_{21}&q_{2}p_{1}-p_{2}-\lambda p_{1}\end{array}\right)$ (25a) where
$a_{21}=\frac{1}{8}\left(\lambda^{3}-q_{2}\lambda^{2}-q_{1}\lambda+\frac{3}{2}q_{1}q_{2}+\frac{5}{8}q_{2}^{3}-8\alpha_{2}\right)\left(8\lambda^{2}+4q_{2}\lambda+4q_{1}+3q_{2}^{2}\right)-\lambda\frac{\partial
h^{(q)}}{\partial q_{1}}-\frac{\partial f^{(q)}}{\partial q_{1}}.$ Here
$f^{(q)}$ is given by $\displaystyle f^{(q)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(4q_{1}-5q_{2}^{2}\right)p_{1}^{2}+4q_{2}p_{1}p_{2}-2p_{2}^{2}+\frac{1}{4}\alpha_{0}\left(4q_{1}+3q_{2}^{2}\right)-\frac{1}{2}\alpha_{1}q_{2}\left(4q_{1}+3q_{2}^{2}\right)$
(25b) $\displaystyle\hskip
42.67912pt-\frac{1}{4}\alpha_{2}(4q_{1}+q_{2}^{2})(4q_{1}+3q_{2}^{2})+\frac{1}{512}q_{2}(4q_{1}+q_{2}^{2})(4q_{1}+3q_{2}^{2})(12q_{1}+7q_{2}^{2}),$
which is a first integral, appearing in the characteristic equation of
$L^{(2)}$:
$z^{2}+\frac{1}{2}f^{(q)}+\frac{1}{2}h^{(q)}\lambda+\alpha_{0}\lambda^{2}+4\alpha_{1}\lambda^{3}+8\alpha_{2}\lambda^{4}-\lambda^{7}=0.$

###### Remark 4.1

The last two terms in $a_{21}$ follow from the formulae:

$u_{2xx}=\left\\{-4p_{1},h^{(q)}\right\\}=4\frac{\partial h^{(q)}}{\partial
q_{1}}\quad\mbox{and}\quad(3u_{2}u_{2x}+2u_{1x})_{x}=8\left\\{q_{2}p_{1}-p_{2},h^{(q)}\right\\}=8\frac{\partial
f^{(q)}}{\partial q_{1}}.$

The Lax representation of the equations generated by $h^{(q)}$ and $f^{(q)}$
are

$L^{(2)}_{x}=\\{L^{(2)},h^{(q)}\\}=[U,L^{(2)}],\quad
L^{(2)}_{t_{1}}=\\{L^{(2)},f^{(q)}\\}=[L^{(1)},L^{(2)}],$ (25c)

where

$U=\begin{pmatrix}0&1\\\
\lambda^{3}-q_{2}\lambda^{2}-q_{1}\lambda+\frac{3}{2}q_{1}q_{2}+\frac{5}{8}q_{2}^{3}-8\alpha_{2}&0\end{pmatrix},\quad
L^{(1)}=\left(\begin{array}[]{cc}p_{1}&\lambda+\frac{1}{2}\,q_{2}\\\
c_{21}&-p_{1}\\\ \end{array}\right),$ (25d)

with
$c_{21}=\left(\lambda^{3}-q_{2}\lambda^{2}-q_{1}\lambda+\frac{3}{2}q_{1}q_{2}+\frac{5}{8}q_{2}^{3}-8\alpha_{2}\right)\left(\lambda+\frac{1}{2}q_{2}\right)-\frac{\partial
h^{(q)}}{\partial q_{1}}$.

##### Using the coordinates of (22c),

we define $L^{(2)}=8V^{(2)}$ to obtain

$\left.\begin{array}[]{l}u_{0}=\frac{1}{16}Q_{1}(8Q_{1}^{2}+3{\bf
Q}^{2})+b_{1},\\\\[2.84526pt] u_{1}=-\frac{1}{8}(6Q_{1}^{2}+{\bf Q}^{2}),\\\
u_{2}=Q_{1},\;\;u_{2x}=P_{1}\\\
u_{1x}=-\frac{1}{4}\left(6Q_{1}P_{1}+\sum_{i=2}^{N}Q_{i}P_{i}\right)\end{array}\right\\}\quad\Rightarrow\quad
L^{(2)}=\left(\begin{array}[]{cc}\sum_{2}^{N}Q_{i}P_{i}-4\lambda
P_{1}&16\lambda^{2}+8\lambda Q_{1}-{\bf Q}^{2}\\\ a_{21}&4\lambda
P_{1}-\sum_{2}^{N}Q_{i}P_{i}\end{array}\right),$ (26a) where
$a_{21}=\left(\lambda^{3}-Q_{1}\lambda^{2}+\frac{1}{8}(6Q_{1}^{2}+{\bf
Q}^{2})\lambda-\frac{1}{16}Q_{1}(8Q_{1}^{2}+3{\bf
Q}^{2})-b_{1}\right)\left(16\lambda^{2}+8Q_{1}\lambda-{\bf
Q}^{2}\right)+4\frac{\partial h^{(Q)}}{\partial Q_{1}}\lambda-\frac{\partial
f^{(Q)}}{\partial Q_{1}},$ (26b) with ${\bf Q}^{2}=\sum_{2}^{N}Q_{i}^{2}$. The
characteristic equation of $L^{(2)}$ is
$z^{2}+2\Phi^{(Q)}+8f^{(Q)}\lambda-32h^{(Q)}\lambda^{2}+64b_{0}\lambda^{3}+256b_{1}\lambda^{4}-256\lambda^{7}=0,$
(26c) where $\Phi^{(Q)}$ and $f^{(Q)}$ are given by $\displaystyle\Phi^{(Q)}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{2\leq i<j\leq
N}h_{ij}+\sum_{i=2}^{N}\beta_{i},$ (26d) $\displaystyle f^{(Q)}$
$\displaystyle=$ $\displaystyle
P_{1}\,\sum_{i=1}^{N}Q_{i}P_{i}-2Q_{1}h^{(Q)}+\frac{1}{2}Q_{1}^{6}+\frac{1}{128}{\bf
Q}^{2}(4Q_{1}^{2}+{\bf Q}^{2})(20Q_{1}^{2}+{\bf Q}^{2})$ (26e)
$\displaystyle\hskip 142.26378pt-\frac{1}{2}b_{0}\left(4Q_{1}^{2}+{\bf
Q}^{2}\right)+2b_{1}Q_{1}\left(2Q_{1}^{2}+{\bf Q}^{2}\right),$ where $h_{ij}$
, given by (16c), are themselves first integrals.

The Lax equations generated by $h^{(Q)},\,f^{(Q)}$ and $h_{ij}$ (not just the
combination $\Phi^{(Q)}$) are

$\displaystyle L^{(2)}_{x}=\\{L^{(2)},h^{(Q)}\\}=[U,L^{(2)}],\qquad
L^{(2)}_{t_{1}}=\\{L^{(2)},f^{(Q)}\\}=[L^{(1)},L^{(2)}],$ (26f)
$\displaystyle\\{L^{(2)},h_{ij}\\}=0,\;\;\;2\leq i<j\leq N,$

where

$L^{(1)}=\left(\begin{array}[]{cc}P_{1}&-4\lambda-2Q_{1}\\\
b_{21}&-P_{1}\end{array}\right),\quad U=\left(\begin{array}[]{cc}0&1\\\
\lambda^{3}-Q_{1}\lambda^{2}+\frac{1}{8}\left(6Q_{1}^{2}+{\bf
Q}^{2}\right)\lambda-\frac{1}{16}Q_{1}(8Q_{1}^{2}+3{\bf
Q}^{2})-b_{1}&0\end{array}\right),$ (26g)

with $b_{21}=\left(\lambda^{3}-Q_{1}\lambda^{2}+\frac{1}{8}(6Q_{1}^{2}+{\bf
Q}^{2})\lambda-\frac{1}{16}Q_{1}(8Q_{1}^{2}+3{\bf
Q}^{2})-b_{1}\right)\left(-4\lambda-2Q_{1}\right)-\frac{\partial
h^{(Q)}}{\partial Q_{1}}.$

##### Using the coordinates of (23),

we have the Lax representation (26a) with
$\sum_{2}^{N}\frac{\beta_{i}}{Q_{i}^{2}}$ replaced by $\frac{\beta}{{\bf
Q}^{2}}$ and the integral (26d) reduces to

$\Phi^{(Q)}=\frac{1}{2}\sum_{2\leq i<j\leq
N}(Q_{i}P_{j}-Q_{j}P_{i})^{2}+\beta,$ (27)

which is just the Casimir ($+\beta$) of the rotation algebra in the
$Q_{2},\dots,Q_{N}$ space.

###  4.3 Bi-Hamiltonian Formulation when $N=2$

Here we have the Hamiltonian of the form

$h^{(Q)}=\frac{1}{2}(P_{1}^{2}+P_{2}^{2})+\frac{\beta_{2}}{Q_{2}^{2}}-b_{0}Q_{1}+\frac{1}{2}b_{1}(4Q_{1}^{2}+Q_{2}^{2})+\frac{1}{64}Q_{1}(4Q_{1}^{2}+Q_{2}^{2})(4Q_{1}^{2}+3Q_{2}^{2}).$
(28)

The Hamiltonians (21c) and (28) define coordinates ${\bf
q}=(q_{i},p_{i},\alpha_{0},\alpha_{1},\alpha_{2})$ and ${\bf
Q}=(Q_{i},P_{i},\beta_{2},b_{0},b_{1})$, respectively. From the definitions of
the canonical coordinates, we have the Poisson map

$\displaystyle q_{1}=-\frac{1}{8}(6Q_{1}^{2}+Q_{2}^{2}),\quad
q_{2}=Q_{1},\;\;p_{1}=-\frac{1}{4}P_{1},\;\;p_{2}=\frac{1}{16}(Q_{2}P_{2}-4Q_{1}P_{1}),$
(29)
$\displaystyle\alpha_{0}=-\frac{1}{8}h^{(Q)},\;\;\alpha_{1}=\frac{1}{16}b_{0},\;\;\alpha_{2}=\frac{1}{8}b_{1},\quad\mbox{together
with}\quad h^{(q)}=\frac{1}{16}f^{(Q)},\;\;f^{(q)}=\frac{1}{64}\beta_{2}.$

###### Remark 4.2

Again, the formulae for $\alpha_{i}$ and $h^{(q)},f^{(q)}$ (in terms of the
variables ${\bf Q}$) can be read directly from the formulae for the
coefficients of the characteristic equations of the Lax matrices in the
respective coordinates.

Again we introduce an extended canonical bracket on each space, labelled
${\cal{P}}_{0}^{(q)}$ and ${\cal{P}}_{1}^{(Q)}$, and use (29) as a Poisson map
to obtain the Poisson tensor ${\cal{P}}_{0}^{(Q)}$ and ${\cal{P}}_{1}^{(q)}$:

${\cal{P}}_{0}^{(q)}=\left(\begin{array}[]{rrrrrrr}0&0&1&0&0&0&0\\\
0&0&0&1&0&0&0\\\ -1&0&0&0&0&0&0\\\ 0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right)\quad\Rightarrow\quad{\cal{P}}_{0}^{(Q)}=\left(\begin{array}[]{ccccccc}0&0&0&\frac{1}{Q_{2}}&a_{15}&0&0\\\\[5.69054pt]
0&0&\frac{1}{Q_{2}}&-\frac{2Q_{1}}{Q_{2}^{2}}&a_{25}&0&0\\\\[5.69054pt]
0&-\frac{1}{Q_{2}}&0&\frac{P_{2}}{Q_{2}^{2}}&a_{35}&0&0\\\\[5.69054pt]
-\frac{1}{Q_{2}}&\frac{2Q_{1}}{Q_{2}^{2}}&-\frac{P_{2}}{Q_{2}^{2}}&0&a_{45}&0&0\\\\[5.69054pt]
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right),$ (30a) where the column
$(a_{15},a_{25},a_{35},a_{45},0,0,0)^{T}=-P_{1}^{(Q)}\nabla_{Q}f^{(Q)}$, with
${\cal{P}}_{1}^{(Q)}$ being the compatible canonical bracket, from which we
build ${\cal{P}}_{1}^{(q)}$:
${\cal{P}}_{1}^{(q)}=\left(\begin{array}[]{ccccccc}0&0&12q_{2}&15q_{2}^{2}+4q_{1}&b_{15}&0&0\\\
0&0&-8&-8q_{2}&b_{25}&0&0\\\ -12q_{2}&8&0&8p_{1}&b_{35}&0&0\\\
-15q_{2}^{2}-4q_{1}&8q_{2}&-8p_{1}&0&b_{45}&0&0\\\
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right),$ (30b) in which the column
$(b_{15},b_{25},b_{35},b_{45},0,0,0)^{T}=-4P_{0}^{(q)}\nabla_{q}h^{(q)}$.

In the ${\bf q}$ space, the bi-Hamiltonian ladder is

$\displaystyle{\bf q}_{t_{h}}={\cal P}_{0}^{(q)}\nabla_{q}h^{(q)}={\cal
P}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{4}\alpha_{0}\right),\quad{\bf
q}_{t_{f}}={\cal P}_{0}^{(q)}\nabla_{q}f^{(q)}={\cal
P}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{8}h^{(q)}\right),$ (31a)
$\displaystyle{\cal P}_{0}^{(q)}\nabla_{q}\alpha_{0}={\cal
P}_{0}^{(q)}\nabla_{q}\alpha_{1}={\cal P}_{0}^{(q)}\nabla_{q}\alpha_{2}={\cal
P}_{1}^{(q)}\nabla_{q}f^{(q)}={\cal P}_{1}^{(q)}\nabla_{q}\alpha_{1}={\cal
P}_{1}^{(q)}\nabla_{q}\alpha_{2}=0,$ and in $\bf{Q}$ space we have
$\displaystyle{\bf Q}_{t_{h}}={\cal P}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\cal
P}_{0}^{(Q)}\nabla_{Q}f^{(Q)},\quad{\bf Q}_{t_{f}}={\cal
P}_{1}^{(Q)}\nabla_{Q}f^{(Q)}={\cal P}_{0}^{(Q)}\nabla_{Q}(-\beta_{2}),$ (31b)
$\displaystyle{\cal P}_{1}^{(Q)}\nabla_{Q}b_{0}={\cal
P}_{1}^{(Q)}\nabla_{Q}b_{1}={\cal P}_{1}^{(Q)}\nabla_{Q}\beta_{2}={\cal
P}_{0}^{(Q)}\nabla_{Q}b_{0}={\cal P}_{0}^{(Q)}\nabla_{Q}b_{1}={\cal
P}_{0}^{(Q)}\nabla_{Q}h^{(Q)}=0.$

Here $f^{(q)}$ and $f^{(Q)}$ are given in (25b) and (26e) (with $N=2$)
respectively.

##  5 The Stationary $t_{1}$ Flow in Quadri-Hamiltonian Form

We now use the Miura maps of Section 2.1.1 to construct a quadri-Hamiltonian
formulation of the stationary flows corresponding to (4b) and (6). In this
way, Figure 1 is extended to an array of 16 Poisson matrices ${\cal
P}_{k}^{(m)},\;k,m=0,1,2,3$. Here the index “$m$” refers to the particular
space, with the correspondence $(0,1,2,3)=\left({\bf
q},{\bf\hat{Q}},{\bf\bar{Q}},{\bf\tilde{Q}}\right)$.

###  5.1 Defining the Coordinates

The 4 Poisson matrices ${\cal P}_{k}^{(k)}$ can be directly constructed in
canonical form, giving us coordinates $(q_{i},p_{i})$,
$(\hat{Q}_{i},\hat{P}_{i})$, $(\bar{Q}_{i},\bar{P}_{i})$ and
$(\tilde{Q}_{i},\tilde{P}_{i})$ respectively.

##### Defining the $u$ space, using $B_{0}^{u}$:

We previously defined these canonical coordinates when deriving (11b), to
obtain

$h^{(q)}=\frac{1}{2}p_{1}^{2}+(4\alpha_{2}^{2}-\alpha_{0})q_{1}+2\alpha_{1}q_{1}^{2}-2\alpha_{2}q_{1}^{3}+\frac{1}{4}q_{1}^{5}.$
(32)

##### Defining the $w$ space, using $B_{1}^{w}$:

The stationary $t_{n}$ flow (from (6)) is defined by

$B_{1}^{w}\delta_{w}H_{n+2}^{w}=0\quad\Rightarrow\quad\delta_{w}\left(H_{n+2}^{w}-\hat{\beta}_{0}w_{0}-\hat{\beta}_{1}w_{2}-\hat{\beta}_{2}\left(4w_{1}+w_{2}^{2}\right)\right)=0,$
(33a) which gives ${\cal
L}_{n+2}^{w}=H_{n+2}^{w}-\hat{\beta}_{0}w_{0}-\hat{\beta}_{1}w_{2}-\hat{\beta}_{2}\left(4w_{1}+w_{2}^{2}\right).$
(33b) For the $t_{1}$ flow, we use $4H_{3}^{w}$, giving ${\cal
L}_{3}^{w}=-2(w_{0x}+w_{0}^{2})w_{2}+w_{1}^{2}+\frac{3}{2}w_{1}w_{2}^{2}+\frac{5}{16}w_{2}^{4}-\hat{\beta}_{0}w_{0}-\hat{\beta}_{1}w_{2}-\hat{\beta}_{2}(4w_{1}+w_{2}^{2}).$
(33c) This is degenerate and leads to
$w_{0}=\frac{2w_{2x}-\hat{\beta}_{0}}{4w_{2}}$ and
$w_{1}=-\frac{3}{4}w_{2}^{2}+2\hat{\beta}_{2}$. Removing an exact derivative,
we find ${\cal
L}_{3}^{w}=\frac{(w_{2x}-\frac{1}{2}\hat{\beta}_{0})^{2}}{2w_{2}}+\frac{1}{4}w_{2}(8\hat{\beta}_{2}w_{2}-w_{2}^{3}-4\hat{\beta}_{1}),$
(33d) corresponding to the Hamiltonian
$h^{(\hat{Q})}=\frac{1}{2}\hat{Q}_{1}\hat{P}_{1}^{2}+\frac{1}{2}\hat{\beta}_{0}\hat{P}_{1}+\frac{1}{4}\hat{Q}_{1}(\hat{Q}_{1}^{3}-8\hat{\beta}_{2}\hat{Q}_{1}+4\hat{\beta}_{1}),$
(33e) where
$\hat{Q}_{1}=w_{2},\,\hat{P}_{1}=\frac{w_{2x}-\frac{1}{2}\hat{\beta}_{0}}{w_{2}}$.

##### Defining the $v$ space, using $B_{2}^{v}$:

The stationary $t_{n}$ flow (from (6)) is defined by

$B_{2}^{v}\delta_{v}H_{n+1}^{v}=0\;\;\Rightarrow\;\;\delta_{v}\left(H_{n+1}^{v}-\gamma_{0}v_{0}-\gamma_{1}v_{1}-\gamma_{2}v_{2}\right)=0,$
(34a) which gives ${\cal
L}_{n+1}^{v}=H_{n+1}^{v}-\gamma_{0}v_{0}-\gamma_{1}v_{1}-\gamma_{2}v_{2}.$
(34b) For the $t_{1}$ flow, we use $-H_{2}^{v}$, removing an exact derivative,
to obtain ${\cal
L}_{2}^{v}=\frac{1}{2}v_{1x}v_{2}+v_{0}^{2}+v_{0}v_{1}v_{2}-\frac{1}{8}v_{2}^{3}-\gamma_{0}v_{0}-\gamma_{1}v_{1}-\gamma_{2}v_{2}.$
(34c) This is degenerate and leads to
$v_{0}=\frac{v_{2x}+2\gamma_{1}}{2v_{2}}$ and
$v_{1}=\frac{\gamma_{0}v_{2}-v_{2x}-2\gamma_{1}}{v_{2}^{2}}$. Removing an
exact derivative, we find ${\cal
L}_{2}^{v}=\frac{(v_{2x}-\gamma_{0}v_{2})^{2}}{4v_{2}^{2}}-\frac{v_{2}^{3}}{8}+\frac{\gamma_{1}^{2}}{v_{2}^{2}}-\frac{\gamma_{0}\gamma_{1}}{v_{2}}-\gamma_{2}v_{2},$
(34d) which gives the Hamiltonian
$h^{(\bar{Q})}=\bar{Q}_{1}^{2}\bar{P}_{1}^{2}+\gamma_{0}\bar{Q}_{1}\bar{P}_{1}+\frac{\bar{Q}_{1}^{3}}{8}+\frac{\gamma_{0}\gamma_{1}}{\bar{Q}_{1}}-\frac{\gamma_{1}^{2}}{\bar{Q}_{1}^{2}}+\gamma_{2}\bar{Q}_{1},$
(34e) where
$\bar{Q}_{1}=v_{2},\,\bar{P}_{1}=\frac{v_{2x}-\gamma_{0}v_{2}}{2v_{2}^{2}}$.

##### Defining the $z$ space, using $B_{3}^{z}$:

The stationary $t_{n}$ flow (from (6)) is defined by

$B_{3}^{z}\delta_{z}H_{n}^{z}=0\quad\Rightarrow\quad\delta_{z}\left(H_{n}^{z}-\mu_{0}z_{0}-\mu_{1}z_{1}-\mu_{2}z_{2}\right)=0,$
(35a) which gives ${\cal
L}_{n}^{z}=H_{n}^{z}-\mu_{0}z_{0}-\mu_{1}z_{1}-\mu_{2}z_{2}.$ (35b) For the
$t_{1}$ flow, we use $H_{1}^{z}$, removing an exact derivative, to obtain
${\cal
L}_{1}^{z}=\frac{1}{4}(z_{1}^{2}+z_{2x})^{2}+z_{0}z_{2}z_{2x}+z_{0}z_{1}(z_{1}z_{2}-2)+z_{0}^{2}z_{2}^{2}-\mu_{0}z_{0}-\mu_{1}z_{1}-\mu_{2}z_{2}.$
(35c) This is degenerate and leads to
$z_{0}=\frac{1}{36z_{2}^{3}}\left((\mu_{0}z_{2}+4)\sigma-12z_{2}^{2}z_{2x}+\mu_{0}z_{2}(10-\mu_{0}z_{2})-6\mu_{1}z_{2}^{3}+8\right),\quad
z_{1}=\frac{\sigma-\mu_{0}z_{2}+2}{6z_{2}},$ (35d) with
$\sigma=\sqrt{12\mu_{1}z_{2}^{3}+(\mu_{0}^{2}-12z_{2x})z_{2}^{2}+8\mu_{0}z_{2}+4}$.
The Lagrangian is then written ${\cal
L}_{1}^{z}=-\frac{\sigma^{3}+8}{108z_{2}^{4}}-\frac{2\mu_{0}}{9z_{2}^{3}}+\frac{12(\mu_{0}z_{2}+1)z_{2x}-5\mu_{0}^{2}}{36z_{2}^{2}}+\frac{\mu_{0}^{3}-36\mu_{1}}{108z_{2}}-\mu_{2}z_{2}-\frac{1}{3}\mu_{0}\mu_{1},$
(35e) leading to canonical variables
$\tilde{Q}_{1}=z_{2},\;\tilde{P}_{1}=\frac{1}{6z_{2}^{2}}\left(\sigma+2\mu_{0}z_{2}+2\right)$
and Hamiltonian
$h^{(\tilde{Q})}=-\tilde{Q}_{1}^{2}\tilde{P}_{1}^{3}+(\mu_{0}\tilde{Q}_{1}+1)\tilde{P}_{1}^{2}+\frac{1}{4}(4\mu_{1}\tilde{Q}_{1}-\mu_{0}^{2})\tilde{P}_{1}+\mu_{2}\tilde{Q}_{1}.$
(35f)

###  5.2 The Miura Maps in these Coordinates

We consider the three steps induced by ${\bf w}\mapsto{\bf u}$, ${\bf
v}\mapsto{\bf w}$ and ${\bf z}\mapsto{\bf v}$. We again extend each space to
include the parameters as dynamical variables, which we define as ${\bf
q}=(q_{1},p_{1},\alpha_{0},\alpha_{1},\alpha_{2})$,
${\bf\hat{Q}}=(\hat{Q}_{1},\hat{P}_{1},\hat{\beta}_{0},\hat{\beta}_{1},\hat{\beta}_{2})$,
${\bf\bar{Q}}=(\bar{Q}_{1},\bar{P}_{1},\gamma_{0},\gamma_{1},\gamma_{2})$ and
${\bf\tilde{Q}}=(\tilde{Q}_{1},\tilde{P}_{1},\mu_{0},\mu_{1},\mu_{2})$.

##### The relation of $\bf q$ to $\bf\hat{Q}$

is given by

$q_{1}=\hat{Q}_{1},\;\;p_{1}=\hat{Q}_{1}\hat{P}_{1}+\frac{\hat{\beta}_{0}}{2},\;\;\alpha_{0}=h^{(\hat{Q})}+4\hat{\beta}_{2}^{2},\;\;\alpha_{1}=\frac{1}{2}\hat{\beta}_{1},\;\;\alpha_{2}=\hat{\beta}_{2},$
(36)

leading to $h^{(q)}=\frac{1}{8}\hat{\beta}_{0}^{2}$.

##### The relation of $\bf\hat{Q}$ to $\bf\bar{Q}$

is given by

$\hat{Q}_{1}=\bar{Q}_{1},\;\;\hat{P}_{1}=2\bar{Q}_{1}\bar{P}_{1}\\!+\\!\frac{2\gamma_{1}}{\bar{Q}_{1}}+\gamma_{0},\;\;\hat{\beta}_{0}=\\!-4\gamma_{1},\;\;\hat{\beta}_{1}=\\!-2h^{(\bar{Q})}\\!-\frac{1}{2}\gamma_{0}^{2},\;\;\hat{\beta}_{2}=-\gamma_{2},$
(37)

leading to $h^{(\hat{Q})}=-2\gamma_{0}\gamma_{1}$.

##### The relation of $\bf\bar{Q}$ to $\bf\tilde{Q}$

is given by

$\bar{Q}_{1}=-2\tilde{P}_{1},\;\;\bar{P}_{1}=-\frac{1}{2}\tilde{Q}_{1}\tilde{P}_{1}+\frac{\mu_{2}}{4\tilde{P}_{1}^{2}}+\frac{1}{4}\mu_{0},\;\;\gamma_{0}=-\mu_{1},\;\;\gamma_{1}=-\mu_{2},\quad\gamma_{2}=-\frac{1}{2}h^{(\tilde{Q})}+\frac{1}{4}\mu_{0}\mu_{1},$
(38)

leading to $h^{(\bar{Q})}=\frac{1}{2}\mu_{0}\mu_{2}$.

###  5.3 The Array of Poisson Brackets

We constructed the 4 canonical representations (32), (33e), (34e) and (35f),
respectively on spaces $\bf q$, $\bf\hat{Q}$, $\bf\bar{Q}$ and $\bf\tilde{Q}$.
The canonical brackets are then extended to include the parameters as
Casimirs:

${\cal{P}}^{(q)}_{0}={\cal{P}}^{(\hat{Q})}_{1}={\cal{P}}^{(\bar{Q})}_{2}={\cal{P}}^{(\tilde{Q})}_{3}=\left(\begin{array}[]{rrrrr}0&1&0&0&0\\\
-1&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right),$ (39)

from which we construct the other 12 Poisson brackets, using the Miura maps of
Section 5.2 as Poisson maps.

##### In the ${\bf q}$ space,

(with $h^{(q)}$ given by (32)) we have

$\displaystyle{\cal{P}}^{(q)}_{1}=\begin{pmatrix}0&a_{12}&a_{13}&0&0\\\
-a_{12}&0&a_{23}&0&0\\\ -a_{13}&-a_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{pmatrix},\quad{\cal{P}}^{(q)}_{2}=\begin{pmatrix}0&b_{12}&0&b_{14}&0\\\
-b_{12}&0&0&b_{24}&0\\\ 0&0&0&0&0\\\ -b_{14}&-b_{24}&0&0&0\\\
0&0&0&0&0\end{pmatrix},$ (40a)
$\displaystyle{\cal{P}}^{(q)}_{3}=\begin{pmatrix}0&c_{12}&8\alpha_{2}c_{15}&0&c_{15}\\\
-c_{12}&0&8\alpha_{2}c_{25}&0&c_{25}\\\
-8\alpha_{2}c_{15}&-8\alpha_{2}c_{25}&0&0&0\\\ 0&0&0&0&0\\\
-c_{15}&-c_{25}&0&0&0\end{pmatrix},$ where
$(a_{13},a_{23},0,0,0)^{T}=(b_{14},b_{24},0,0,0)^{T}=(c_{15},c_{25},0,0,0)^{T}={\cal{P}}^{(q)}_{0}\nabla_{q}h^{(q)},\,a_{12}=-h^{(q)}_{\alpha_{0}},\,b_{12}=-h^{(q)}_{\alpha_{1}}$,
$c_{12}=-\left(\partial_{\alpha_{2}}+8\alpha_{2}\partial_{\alpha_{0}}\right)h^{(q)}$.

Each of these has 3 Casimirs and the $t_{1}=t_{h}$ flow has a quadri-
Hamiltonian representation:

$\displaystyle{\bf
q}_{t_{h}}={\cal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{1}^{(q)}\nabla_{q}\alpha_{0}={\cal{P}}_{2}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{3}^{(q)}\nabla_{q}\alpha_{2},$
$\displaystyle{\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}={\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{2}={\cal{P}}_{1}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{1}^{(q)}\nabla_{q}\alpha_{2}$
(40b)
$\displaystyle={\cal{P}}_{2}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{2}^{(q)}\nabla_{q}\alpha_{0}={\cal{P}}_{2}^{(q)}\nabla_{q}\alpha_{2}={\cal{P}}_{3}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{3}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{3}^{(q)}\nabla_{q}(\alpha_{0}-4\alpha_{2}^{2})=0.$

###### Remark 5.1

The first 3 rows and columns of ${\cal{P}}^{(q)}_{1}$ are in the form of a
standard $3\times 3$ Poisson bracket, with Casimir
$h^{(q)}(q_{1},p_{1},\alpha_{0})$. The form of ${\cal{P}}^{(q)}_{3}$ shows how
this is generalised, as will be seen in further examples below.

##### In the ${\bf\hat{Q}}$ space,

(with $h^{(\hat{Q})}$ given by (33e)) we have

$\displaystyle{\cal{P}}^{(\hat{Q})}_{0}=\frac{1}{\hat{\beta}_{0}}\left(\begin{array}[]{ccccc}0&a_{12}&a_{13}&0&0\\\
-a_{12}&0&a_{23}&0&0\\\ -a_{13}&-a_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),\quad{\cal{P}}^{(\hat{Q})}_{2}=\begin{pmatrix}0&b_{12}&0&b_{14}&0\\\
-b_{12}&0&0&b_{24}&0\\\ 0&0&0&0&0\\\ -b_{14}&-b_{24}&0&0&0\\\
0&0&0&0&0\end{pmatrix},$ (41f) (41g)
$\displaystyle{\cal{P}}^{(\hat{Q})}_{3}=\begin{pmatrix}0&c_{12}&0&0&c_{15}\\\
-c_{12}&0&0&0&c_{25}\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
-c_{15}&-c_{25}&0&0&0\end{pmatrix},$ where
$(a_{13},a_{23},0,0,0)^{T}=(b_{14},b_{24},0,0,0)^{T}=(c_{15},c_{25},0,0,0)^{T}={\cal{P}}^{(\hat{Q})}_{1}\nabla_{\hat{Q}}h^{(\hat{Q})},\,a_{12}=-h^{(\hat{Q})}_{\hat{\beta}_{0}},\,b_{12}=-h^{(\hat{Q})}_{\hat{\beta}_{1}}$,
$c_{12}=-h^{(\hat{Q})}_{\hat{\beta}_{2}}$.

Each of these has 3 Casimirs and the $t_{1}=t_{h}$ flow has a quadri-
Hamiltonian representation:

$\displaystyle{\bf\hat{Q}}_{t_{h}}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\left(\frac{1}{2}\hat{\beta}_{0}^{2}\right)={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2},$
$\displaystyle{\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}$
(41h)
$\displaystyle={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}=0.$

##### In the ${\bf\bar{Q}}$ space,

(with $h^{(\bar{Q})}$ given by (34e)) we have

$\displaystyle{\cal{P}}^{(\bar{Q})}_{0}=\frac{1}{\gamma_{1}^{2}}\left(\begin{array}[]{ccccc}0&a_{12}&-\gamma_{0}a_{13}&\gamma_{1}a_{13}&0\\\
-a_{12}&0&-\gamma_{0}a_{23}&\gamma_{1}a_{23}&0\\\
\gamma_{0}a_{13}&\gamma_{0}a_{23}&0&0&0\\\
-\gamma_{1}a_{13}&-\gamma_{1}a_{23}&0&0&0\\\
0&0&0&0&0\end{array}\right),\quad{\cal{P}}^{(\bar{Q})}_{1}=\frac{1}{\gamma_{1}}\begin{pmatrix}0&b_{12}&b_{13}&0&0\\\
-b_{12}&0&b_{23}&0&0\\\ -b_{13}&-b_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{pmatrix},$ (42f) (42g)
$\displaystyle{\cal{P}}^{(\bar{Q})}_{3}=\begin{pmatrix}0&c_{12}&0&0&c_{15}\\\
-c_{12}&0&0&0&c_{25}\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
-c_{15}&-c_{25}&0&0&0\end{pmatrix},$ where
$(a_{13},a_{23},0,0,0)^{T}=(b_{13},b_{23},0,0,0)^{T}=(c_{15},c_{25},0,0,0)^{T}={\cal
P}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}$ and

$a_{12}=\left(\gamma_{0}\partial_{\gamma_{0}}-\gamma_{1}\partial_{\gamma_{1}}\right)\left(h^{(\bar{Q})}+\frac{1}{4}\gamma_{0}^{2}\right),\,b_{12}=-\partial_{\gamma_{0}}\left(h^{(\bar{Q})}+\frac{1}{4}\gamma_{0}^{2}\right),\,c_{12}=-h^{(\bar{Q})}_{\gamma_{2}}$.

Each of these has 3 Casimirs and the $t_{1}=t_{h}$ flow has a quadri-
Hamiltonian representation:

$\displaystyle{\bf\bar{Q}}_{t_{h}}={\cal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}={\cal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(\frac{1}{2}\gamma_{1}^{2}\right)={\cal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}(\gamma_{0}\gamma_{1})={\cal{P}}_{3}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{2},$
$\displaystyle{\cal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}={\cal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\cal{P}}_{2}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{2}={\cal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}(\gamma_{0}\gamma_{1})={\cal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{2}={\cal{P}}_{0}^{(\bar{Q})}\nabla_{\bar{Q}}\left(h^{(\bar{Q})}+\frac{1}{4}\gamma_{0}^{2}\right)$
(42h)
$\displaystyle={\cal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\cal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{2}={\cal{P}}_{1}^{(\bar{Q})}\nabla_{\bar{Q}}\left(h^{(\bar{Q})}+\frac{1}{4}\gamma_{0}^{2}\right)={\cal{P}}_{3}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{0}={\cal{P}}_{3}^{(\bar{Q})}\nabla_{\bar{Q}}\gamma_{1}={\cal{P}}_{3}^{(\bar{Q})}\nabla_{\bar{Q}}h^{(\bar{Q})}=0.$

##### In the ${\bf\tilde{Q}}$ space,

(with $h^{(\tilde{Q})}$ given by (35f)) we have

$\displaystyle{\cal{P}}^{(\tilde{Q})}_{0}=\frac{1}{\mu_{2}^{3}}\left(\begin{array}[]{ccccc}0&a_{12}&(\mu_{1}^{2}-\mu_{0}\mu_{2})a_{13}&-\mu_{1}\mu_{2}a_{13}&\mu_{2}^{2}a_{13}\\\
-a_{12}&0&(\mu_{1}^{2}-\mu_{0}\mu_{2})a_{23}&-\mu_{1}\mu_{2}a_{23}&\mu_{2}^{2}a_{23}\\\
(\mu_{0}\mu_{2}-\mu_{1}^{2})a_{13}&(\mu_{0}\mu_{2}-\mu_{1}^{2})a_{23}&0&0&0\\\
\mu_{1}\mu_{2}a_{13}&\mu_{1}\mu_{2}a_{23}&0&0&0\\\
-\mu_{2}^{2}a_{13}&-\mu_{2}^{2}a_{23}&0&0&0\end{array}\right),$ (43f) (43g)
$\displaystyle{\cal{P}}^{(\tilde{Q})}_{1}=\frac{1}{\mu_{2}^{2}}\begin{pmatrix}0&b_{12}&-\mu_{1}b_{13}&\mu_{2}b_{13}&0\\\
-b_{12}&0&-\mu_{1}b_{23}&\mu_{2}b_{23}&0\\\
\mu_{1}b_{13}&\mu_{1}b_{23}&0&0&0\\\ -\mu_{2}b_{13}&-\mu_{2}b_{23}&0&0&0\\\
0&0&0&0&0\end{pmatrix},\quad{\cal{P}}^{(\tilde{Q})}_{2}=\frac{1}{\mu_{2}}\begin{pmatrix}0&c_{12}&c_{13}&0&0\\\
-c_{12}&0&c_{23}&0&0\\\ -c_{13}&-c_{23}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{pmatrix},$ where
$(a_{13},a_{23},0,0,0)^{T}=(b_{13},b_{23},0,0,0)^{T}=(c_{13},c_{23},0,0,0)^{T}={\cal
P}_{3}^{(\tilde{Q})}\nabla_{\tilde{Q}}h^{(\tilde{Q})}$ and

$a_{12}=\left((\mu_{0}\mu_{2}-\mu_{1}^{2})\partial_{\mu_{0}}+\mu_{1}\mu_{2}\partial_{\mu_{1}}-\mu_{2}^{2}\partial_{\mu_{2}}\right)\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right),\,b_{12}=\left(\mu_{1}\partial_{\mu_{0}}-\mu_{2}\partial_{\mu_{1}}\right)\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right)$,
$c_{12}=-\partial_{\mu_{0}}\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right)$.

Each of these has 3 Casimirs and the $t_{1}=t_{h}$ flow has a quadri-
Hamiltonian representation:

$\displaystyle{\bf\tilde{Q}}_{t_{h}}={\cal{P}}_{3}^{(\tilde{Q})}\nabla_{\tilde{Q}}h^{(\tilde{Q})}={\cal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(\frac{1}{2}\mu_{2}^{2}\right)={\cal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}(\mu_{1}\mu_{2})={\cal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}(\mu_{0}\mu_{2}),$
$\displaystyle{\cal{P}}_{3}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{0}={\cal{P}}_{3}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{1}={\cal{P}}_{3}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{2}={\cal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}(\mu_{1}\mu_{2})={\cal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(2\mu_{0}\mu_{2}+\mu_{1}^{2}\right)$
(43h)
$\displaystyle={\cal{P}}_{0}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right)={\cal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{2}={\cal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(2\mu_{0}\mu_{2}+\mu_{1}^{2}\right)={\cal{P}}_{1}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right)$
$\displaystyle\qquad={\cal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{1}={\cal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}\mu_{2}={\cal{P}}_{2}^{(\tilde{Q})}\nabla_{\tilde{Q}}\left(h^{(\tilde{Q})}-\frac{1}{2}\mu_{0}\mu_{1}\right)=0.$

###### Remark 5.2

When $M=3$, we have all 4 spaces, each 5 dimensional and with 4 Poisson
brackets.

When $M=2$ (the DWW case), we no longer have the $\tilde{\bf Q}$ space and for
each of the other spaces we reduce to 4 dimensions (losing parameters
$\alpha_{2},\hat{\beta}_{2},\gamma_{2}$). For each space we only have the
first 3 Poisson brackets, which can be written as $4\times 4$ matrices.

When $M=1$ (the KdV case), we no longer have the $\bar{\bf Q}$ space, with the
$\hat{\bf Q}$ and $\bf q$ spaces reducing to 3 dimensions (losing parameters
$\alpha_{1},\hat{\beta}_{1}$). Each of these spaces only has the first 2
Poisson brackets, which can be written as $3\times 3$ matrices.

######  Remark 5.3 (Quadri-Hamiltonian formulation of (17))

Since we have 4 Poisson brackets in the $\bf q$ space, we can use the Poisson
map (18) to derive 2 further Poisson brackets for (17), in the $\bf Q$ space.
We don’t explicitly present this for the $t_{1}$ flow, but the corresponding
results for the $t_{2}$ flow are given in Section 6.4.

##  6 The Stationary $t_{2}$ Flow in Quadri-Hamiltonian Form

Following the approach outlined in Section 5, we now consider the $t_{2}$
flow, with the same definition of ${\cal{P}}_{k}^{(m)}$.

###  6.1 Defining the Coordinates

The 4 Poisson matrices ${\cal P}_{k}^{(k)}$ can be directly constructed in
canonical form, giving us coordinates $(q_{i},p_{i})$,
$(\hat{Q}_{i},\hat{P}_{i})$, $(\bar{Q}_{i},\bar{P}_{i})$ and
$(\tilde{Q}_{i},\tilde{P}_{i})$ respectively.

##### Defining the $u$ space, using $B_{0}^{u}$:

We previously defined these canonical coordinates when deriving (21c), giving

$h^{(q)}=5q_{2}p_{1}^{2}-4p_{1}p_{2}+\alpha_{0}q_{2}+\alpha_{1}(4q_{1}+q_{2}^{2})-4\alpha_{2}q_{2}(2q_{1}+q_{2}^{2})-\frac{1}{4}q_{1}^{3}+\frac{3}{16}q_{1}^{2}q_{2}^{2}+\frac{25}{64}q_{1}q_{2}^{4}+\frac{29}{256}q_{2}^{6}.$
(44)

##### Defining the $w$ space, using $B_{1}^{w}$:

Using (33b), with $n=2$, and choosing $-3H_{4}^{w}$, gives

${\cal
L}_{4}^{w}=\frac{3}{8}(4w_{1}+3w_{2}^{2})(w_{0}^{2}+w_{0x})+\frac{3}{16}w_{2x}^{2}-\frac{3}{128}w_{2}(12w_{1}+7w_{2}^{2})(4w_{1}+w_{2}^{2})-\hat{\beta}_{0}w_{0}-\hat{\beta}_{1}w_{2}-\hat{\beta}_{2}(4w_{1}+w_{2}^{2}).$
(45a) This is degenerate and leads to
$w_{1}=-\frac{5}{12}w_{2}^{2}+\frac{2}{9w_{2}}(3w_{0}^{2}+3w_{0x}-8\hat{\beta}_{2})$,
giving ${\cal
L}_{4}^{w}=\frac{(w_{0}^{2}+w_{0x})^{2}}{2w_{2}}+\frac{3w_{2}^{3}-16\hat{\beta}_{2}}{6w_{2}}w_{0x}+\frac{3}{16}w_{2x}^{2}+\frac{1}{32}w_{2}^{2}(w_{2}^{3}+16w_{0}^{2})-\hat{\beta}_{0}w_{0}-\hat{\beta}_{1}w_{2}+\frac{2}{9}\hat{\beta}_{2}\left(3w_{2}^{2}-\frac{12w_{0}^{2}-16\hat{\beta}_{2}}{w_{2}}\right),$
(45b) corresponding to the Hamiltonian
$h^{(\hat{Q})}=\frac{1}{2}\hat{Q}_{2}\hat{P}_{1}^{2}-\frac{1}{6}\left(6\hat{Q}_{1}^{2}+3\hat{Q}_{2}^{3}-16\hat{\beta}_{2}\right)\hat{P}_{1}+\frac{4}{3}\hat{P}_{2}^{2}+\frac{3}{32}\hat{Q}_{2}^{5}+\hat{\beta}_{0}\hat{Q}_{1}+\hat{\beta}_{1}\hat{Q}_{2}-2\hat{\beta}_{2}\hat{Q}_{2}^{2},$
(45c) with
$\hat{Q}_{1}=w_{0},\;\hat{Q}_{2}=w_{2},\;\hat{P}_{1}=\frac{1}{2}w_{2}^{2}+\frac{1}{3w_{2}}(3w_{0}^{2}+3w_{0x}-8\hat{\beta}_{2}),\;\hat{P}_{2}=\frac{3}{8}w_{2x}$.

##### Defining the $v$ space, using $B_{2}^{v}$:

Using (34b), with $n=2$, and choosing $2H_{3}^{v}$ (removing an exact
derivative), we obtain

${\cal
L}_{3}^{v}=\frac{1}{2}(2v_{0}v_{1}+v_{1x})^{2}-\frac{3}{4}v_{2}^{2}(2v_{0}v_{1}+v_{1x})+v_{0}v_{2x}-v_{0}^{2}v_{2}+\frac{5}{32}v_{2}^{4}-\gamma_{0}v_{0}-\gamma_{1}v_{1}-\gamma_{2}v_{2}.$
(46a) This is degenerate and leads to
$v_{0}=\frac{3v_{1}v_{2}^{2}-4v_{1}v_{1x}-2v_{2x}+2\gamma_{0}}{4(2v_{1}^{2}-v_{2})}$.
Removing an exact derivative, we find $\displaystyle{\cal L}_{3}^{v}$
$\displaystyle=$
$\displaystyle\frac{1}{32(2v_{1}^{2}-v_{2})}\left(-16v_{1x}^{2}v_{2}+8(3v_{2}^{3}+4\gamma_{0}v_{1}-4v_{1}v_{2x})v_{1x}-8v_{2x}^{2}+8(3v_{1}v_{2}^{2}+2\gamma_{0})v_{2x}\right.$
(46b) $\displaystyle\hskip
56.9055pt\left.-v_{2}^{4}(8v_{1}^{2}+5v_{2})-8\gamma_{0}(3v_{1}v_{2}^{2}+\gamma_{0})-32(\gamma_{1}v_{1}+\gamma_{2}v_{2})(2v_{1}^{2}-v_{2})\right).$
The Legendre transformation gives coordinates
$\displaystyle\bar{Q}_{1}=v_{1},\quad\bar{Q}_{2}=v_{2},\quad\bar{P}_{1}=\frac{1}{4(2v_{1}^{2}-v_{2})}\left(v_{2}(3v_{2}^{2}-4v_{1x})+4v_{1}(\gamma_{0}-v_{2x})\right),$
(46c)
$\displaystyle\qquad\qquad\quad\bar{P}_{2}=\frac{1}{4(2v_{1}^{2}-v_{2})}\left(v_{1}(3v_{2}^{2}-4v_{1x})-2v_{2x}+2\gamma_{0})\right),$
and Hamiltonian
$h^{(\bar{Q})}=\frac{1}{2}\bar{P}_{1}^{2}-2\bar{Q}_{1}\bar{P}_{1}\bar{P}_{2}+\bar{Q}_{2}\bar{P}_{2}^{2}+\frac{3}{4}\bar{Q}_{2}^{2}\bar{P}_{1}+\gamma_{0}\bar{P}_{2}+\frac{1}{8}\bar{Q}_{2}^{4}+\gamma_{1}\bar{Q}_{1}+\gamma_{2}\bar{Q}_{2}.$
(46d)

##### Defining the $z$ space, using $B_{3}^{z}$:

Using (35b), with $n=2$, and choosing $H_{2}^{z}$ (removing an exact
derivative), we obtain

$\displaystyle{\cal L}_{2}^{z}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}(2z_{0}z_{2}+z_{2x})^{3}-\frac{3}{8}z_{1}^{2}z_{2x}^{2}+\frac{1}{2}z_{1x}z_{2x}-\frac{3}{8}z_{1}^{2}(4z_{0}z_{2}+z_{1}^{2})z_{2x}+z_{0}(z_{2}z_{1x}+z_{1}z_{2x})-z_{0}^{2}$
(47a) $\displaystyle\hskip
56.9055pt-\frac{1}{8}z_{1}^{6}-\frac{1}{4}z_{0}z_{1}(3z_{1}z_{2}-4)(2z_{0}z_{2}+z_{1}^{2})-\mu_{0}z_{0}-\mu_{1}z_{1}-\mu_{2}z_{2}.$
This is degenerate and leads to
$z_{0}=\frac{1}{6z_{2}^{3}}\left(2\sigma-2-3z_{2}^{2}(z_{1}^{2}+z_{2x})+4z_{1}z_{2}\right)$,
with
$\sigma=\sqrt{3z_{2}^{4}z_{1x}-3(z_{1}^{3}+z_{1}z_{2x}+\mu_{0})z_{2}^{3}+(7z_{1}^{2}+3z_{2x})z_{2}^{2}-4z_{1}z_{2}+1},\qquad\mbox{giving}$

${\cal
L}_{2}^{z}=\frac{2(\sigma^{3}-1)}{27z_{2}^{6}}+\frac{4z_{1}}{9z_{2}^{5}}-\frac{11z_{1}^{2}}{9z_{2}^{4}}+\frac{43z_{1}^{3}+9z_{1}z_{2x}+9\mu_{0}}{27z_{2}^{3}}-\frac{11z_{1}^{4}+10z_{1}^{2}z_{2x}+3z_{2x}^{2}+8\mu_{0}z_{1}}{12z_{2}^{2}}+\frac{\mu_{0}}{2}\frac{z_{1}^{2}}{z_{2}}-\mu_{1}z_{1}-\mu_{2}z_{2}.$
(47b)

The Legendre transformation gives coordinates

$\tilde{Q}_{1}=z_{1},\quad\tilde{Q}_{2}=z_{2},\quad\tilde{P}_{1}=\frac{\sigma}{3z_{2}^{2}},\quad\tilde{P}_{2}=\frac{1-z_{1}z_{2}}{3z_{2}^{4}}\sigma+\frac{z_{1}}{3z_{2}^{3}}-\frac{5z_{1}^{2}+3z_{2x}}{6z_{2}^{2}},$
(47c)

and the Hamiltonian

$\displaystyle h^{(\tilde{Q})}$ $\displaystyle=$
$\displaystyle\tilde{P}_{1}^{3}-\frac{1}{\tilde{Q}_{2}^{2}}\left((\tilde{Q}_{1}\tilde{Q}_{2}-1)\tilde{P}_{1}+\tilde{Q}_{2}^{2}\tilde{P}_{2}\right)^{2}+\frac{(3\mu_{0}-2\tilde{Q}_{1}^{3})\tilde{Q}_{2}^{3}+2\tilde{Q}_{1}\tilde{Q}_{2}-1}{3\tilde{Q}_{2}^{4}}\tilde{P}_{1}+\frac{\tilde{Q}_{1}(2-5\tilde{Q}_{1}\tilde{Q}_{2})}{3\tilde{Q}_{2}}\tilde{P}_{2}$
(47d)
$\displaystyle+\frac{2}{27\tilde{Q}_{2}^{6}}\left(3\tilde{Q}_{1}^{4}\tilde{Q}_{2}^{4}-14\tilde{Q}_{1}^{3}\tilde{Q}_{2}^{3}+15\tilde{Q}_{1}^{2}\tilde{Q}_{2}^{2}-6\tilde{Q}_{1}\tilde{Q}_{2}+1\right)+\frac{\mu_{0}}{6}\frac{4\tilde{Q}_{1}\tilde{Q}_{2}-3\tilde{Q}_{1}^{2}\tilde{Q}_{2}^{2}-2}{\tilde{Q}_{2}^{3}}+\mu_{1}\tilde{Q}_{1}+\mu_{2}\tilde{Q}_{2}.$

###  6.2 The Miura Maps in these Coordinates

We consider the three steps induced by ${\bf w}\mapsto{\bf u}$, ${\bf
v}\mapsto{\bf w}$ and ${\bf z}\mapsto{\bf v}$. We again extend each space to
include the parameters as dynamical variables, which we define as ${\bf
q}=(q_{i},p_{i},\alpha_{0},\alpha_{1},\alpha_{2})$,
${\bf\hat{Q}}=(\hat{Q}_{i},\hat{P}_{i},\hat{\beta}_{0},\hat{\beta}_{1},\hat{\beta}_{2})$,
${\bf\bar{Q}}=(\bar{Q}_{i},\bar{P}_{i},\gamma_{0},\gamma_{1},\gamma_{2})$ and
${\bf\tilde{Q}}=(\tilde{Q}_{i},\tilde{P}_{i},\mu_{0},\mu_{1},\mu_{2})$.

##### The relation of $\bf q$ to $\bf\hat{Q}$

is given by

$\begin{split}&q_{1}=-\frac{3}{4}\hat{Q}_{2}^{2}+\frac{2}{3}\hat{P}_{1},\quad
q_{2}=\hat{Q}_{2},\quad p_{1}=-\frac{2}{3}\hat{P}_{2},\quad
p_{2}=-\frac{1}{3}(\hat{Q}_{1}\hat{P}_{1}+2\hat{Q}_{2}\hat{P}_{2})+\frac{\hat{\beta}_{0}}{6},\\\
&\quad\quad\alpha_{0}=-\frac{1}{3}h^{(\hat{Q})},\quad\alpha_{1}=-\frac{1}{6}\hat{\beta}_{1},\quad\alpha_{2}=-\frac{1}{3}\hat{\beta}_{2},\end{split}$
(48a) leading to
$h^{(q)}=-\frac{1}{9}f^{(\hat{Q})},\;\;f^{(q)}=-\frac{1}{18}\hat{\beta}_{0}^{2}$,
where $f^{(q)}$ is given by (25b) and
$f^{(\hat{Q})}=\frac{2}{3}\hat{P}_{1}^{3}-\frac{3}{2}\hat{Q}_{2}^{2}\hat{P}_{1}^{2}+8\hat{Q}_{1}\hat{P}_{1}\hat{P}_{2}+\left(\frac{3}{8}\hat{Q}_{2}^{4}-3\hat{Q}_{1}^{2}\hat{Q}_{2}-8\hat{\beta}_{2}\hat{Q}_{2}+4\hat{\beta}_{1}\right)\hat{P}_{1}-\hat{\beta}_{0}(4\hat{P}_{2}-3\hat{Q}_{1}\hat{Q}_{2}).$
(48b)

##### The relation of $\bf\hat{Q}$ to $\bf\bar{Q}$

is given by

$\displaystyle\hat{Q}_{1}=\bar{P}_{2},\quad\hat{Q}_{2}=\bar{Q}_{2},\quad\hat{P}_{1}=-\frac{3}{2}\bar{P}_{1},\quad\hat{P}_{2}=\frac{3}{4}(\bar{Q}_{2}\bar{P}_{2}-\bar{Q}_{1}\bar{P}_{1})+\frac{3}{8}\gamma_{0},$
(49a)
$\displaystyle\hat{\beta}_{0}=-\frac{3}{2}\gamma_{1},\quad\hat{\beta}_{1}=-\frac{3}{4}h^{(\bar{Q})},\quad\hat{\beta}_{2}=-\frac{3}{8}\gamma_{2},$
leading to $h^{(\hat{Q})}=\frac{3}{2}f^{(\bar{Q})}+\frac{3}{16}\gamma_{0}^{2}$
and $f^{(\hat{Q})}=\frac{9}{4}\gamma_{0}\gamma_{1}$, where
$f^{(\bar{Q})}={\bar{P}}_{1}{\bar{P}}_{2}^{2}+\frac{1}{2}({\bar{Q}}_{1}^{2}+{\bar{Q}}_{2}){\bar{P}}_{1}^{2}+\frac{1}{8}({\bar{Q}}_{2}^{3}-4\gamma_{0}{\bar{Q}}_{1}+8\gamma_{2}){\bar{P}}_{1}-\gamma_{1}{\bar{P}}_{2}-\frac{\gamma_{1}}{2}{\bar{Q}}_{1}{\bar{Q}}_{2}.$
(49b)

##### The relation of $\bf\bar{Q}$ to $\bf\tilde{Q}$

is given by

$\displaystyle\bar{Q}_{1}=\tilde{Q}_{1},\quad\bar{Q}_{2}=-2\tilde{P}_{1}+\frac{2}{3\tilde{Q}_{2}^{2}}\,\left(1-2\tilde{Q}_{1}\tilde{Q}_{2}\right),\quad\bar{P}_{1}=2\tilde{P}_{2}+\frac{\mu_{0}}{\tilde{Q}_{2}}-\frac{2}{3\tilde{Q}_{2}^{4}}\,\left(\tilde{Q}_{1}\tilde{Q}_{2}-1\right)^{2},$
(50a)
$\displaystyle\bar{P}_{2}=\tilde{Q}_{1}\tilde{P}_{1}+\tilde{Q}_{2}\tilde{P}_{2}+\frac{1}{3\tilde{Q}_{2}^{3}}\,\left(\tilde{Q}_{1}^{2}\tilde{Q}_{2}^{2}+\tilde{Q}_{1}\tilde{Q}_{2}-1\right),\quad\gamma_{0}=2\mu_{1},\quad\gamma_{1}=2\mu_{2},\quad\gamma_{2}=h^{(\tilde{Q})},$
leading to $h^{(\bar{Q})}=f^{(\tilde{Q})}$ and $f^{(\bar{Q})}=\mu_{0}\mu_{2}$,
where $\displaystyle f^{(\tilde{Q})}$ $\displaystyle=$ $\displaystyle
2{\tilde{P}}_{1}^{2}{\tilde{P}}_{2}-\frac{2}{3{\tilde{Q}}_{2}^{4}}({\tilde{Q}}_{1}{\tilde{Q}}_{2}-1)^{2}{\tilde{P}}_{1}^{2}+\frac{2}{9{\tilde{Q}}_{2}^{6}}(3{\tilde{Q}}_{1}^{2}{\tilde{Q}}_{2}^{2}-4{\tilde{Q}}_{1}{\tilde{Q}}_{2}+2)\left(({\tilde{Q}}_{1}{\tilde{Q}}_{2}-1)^{2}-3{\tilde{Q}}_{2}^{4}{\tilde{P}}_{2}\right){\tilde{P}}_{1}$
(50b)
$\displaystyle+\frac{2}{9{\tilde{Q}}_{2}^{4}}(1-2{\tilde{Q}}_{1}{\tilde{Q}}_{2})(9{\tilde{Q}}_{2}^{4}{\tilde{P}}_{2}-3{\tilde{Q}}_{1}^{2}{\tilde{Q}}_{2}^{2}+10{\tilde{Q}}_{1}{\tilde{Q}}_{2}-5){\tilde{P}}_{2}+\frac{4}{27{\tilde{Q}}_{2}^{8}}({\tilde{Q}}_{1}{\tilde{Q}}_{2}-1)^{2}(2{\tilde{Q}}_{1}{\tilde{Q}}_{2}-1)^{2}$
$\displaystyle+\frac{\mu_{0}}{9{\tilde{Q}}_{2}^{5}}\left(9{\tilde{Q}}_{2}^{4}{\tilde{P}}_{1}^{2}-3{\tilde{Q}}_{2}^{2}(3{\tilde{Q}}_{1}^{2}{\tilde{Q}}_{2}^{2}-4{\tilde{Q}}_{1}{\tilde{Q}}_{2}+2){\tilde{P}}_{1}+18(1-{\tilde{Q}}_{1}{\tilde{Q}}_{2}){\tilde{Q}}_{2}^{4}{\tilde{P}}_{2}-11{\tilde{Q}}_{1}^{2}{\tilde{Q}}_{2}^{2}+14{\tilde{Q}}_{1}{\tilde{Q}}_{2}-5\right)$
$\displaystyle\hskip
28.45274pt+\frac{2\mu_{1}}{3{\tilde{Q}}_{2}^{3}}\left(3{\tilde{Q}}_{2}^{4}{\tilde{P}}_{2}-({\tilde{Q}}_{1}{\tilde{Q}}_{2}-1)^{2}\right)+\frac{2\mu_{2}}{3{\tilde{Q}}_{2}}({\tilde{Q}}_{1}{\tilde{Q}}_{2}+1-3{\tilde{Q}}_{2}^{2}{\tilde{P}}_{1})+\frac{\mu_{0}^{2}}{2{\tilde{Q}}_{2}^{2}}.$

###  6.3 The Array of Poisson Brackets

We constructed the 4 canonical representations, with Hamiltonians (44), (45c),
(46d) and (47d), respectively on spaces $\bf q$, $\bf\hat{Q}$, $\bf\bar{Q}$
and $\bf\tilde{Q}$. The canonical brackets are then extended to include the
parameters as Casimirs:

${\cal{P}}^{(q)}_{0}={\cal{P}}^{(\hat{Q})}_{1}={\cal{P}}^{(\bar{Q})}_{2}={\cal{P}}^{(\tilde{Q})}_{3}=\left(\begin{array}[]{rrrrrrr}0&0&1&0&0&0&0\\\
0&0&0&1&0&0&0\\\ -1&0&0&0&0&0&0\\\ 0&-1&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\end{array}\right),$ (51)

from which we can construct the other 12 Poisson brackets. For this paper, we
only present the operators ${\cal P}_{i}^{(q)}$ and ${\cal
P}_{i}^{(\hat{Q})}$.

In the $\bf q$ space, we have ${\cal{P}}^{(q)}_{1}$, given by (30b), and

${\cal{P}}^{(q)}_{2}=\begin{pmatrix}0&0&3q_{2}^{2}-4q_{1}&\frac{1}{2}q_{2}(4q_{1}+15q_{2}^{2})&b_{15}&b_{16}&0\\\
0&0&-4q_{2}&-4q_{1}-7q_{2}^{2}&b_{25}&b_{26}&0\\\
4q_{1}-3q_{2}^{2}&4q_{2}&0&4q_{2}p_{1}&b_{35}&b_{36}&0\\\
-\frac{1}{2}q_{2}(4q_{1}+15q_{2}^{2})&4q_{1}+7q_{2}^{2}&-4q_{2}p_{1}&0&b_{45}&b_{46}&0\\\
-b_{15}&-b_{25}&-b_{35}&-b_{45}&0&0&0\\\
-b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0&0\\\ 0&0&0&0&0&0&0\end{pmatrix},$ (52a)
${\cal{P}}^{(q)}_{3}=\begin{pmatrix}0&0&q_{2}(8q_{1}+3q_{2}^{2})&c_{14}&0&c_{16}&c_{17}\\\
0&0&-4q_{1}-q_{2}^{2}&\frac{1}{2}q_{2}^{3}-2q_{1}q_{2}&0&c_{26}&c_{27}\\\
-q_{2}(8q_{1}+3q_{2}^{2})&4q_{1}+q_{2}^{2}&0&p_{1}(4q_{1}+q_{2}^{2})&0&c_{36}&c_{37}\\\
-c_{14}&2q_{1}q_{2}-\frac{1}{2}q_{2}^{3}&-p_{1}(4q_{1}+q_{2}^{2})&0&0&c_{46}&c_{47}\\\
0&0&0&0&0&0&0\\\ -c_{16}&-c_{26}&-c_{36}&-c_{46}&0&0&0\\\
-c_{17}&-c_{27}&-c_{37}&-c_{47}&0&0&0\end{pmatrix},$ (52b) where
$c_{14}=\frac{1}{8}(4q_{1}+q_{2}^{2})(4q_{1}+15q_{2}^{2})$, and
$\displaystyle\frac{1}{4}(b_{15},b_{25},b_{35},b_{45},0,0,0)^{T}=(c_{16},c_{26},c_{36},c_{46},0,0,0)^{T}={\cal{P}}^{(q)}_{0}\nabla_{q}f^{(q)},$
$\displaystyle(b_{16},b_{26},b_{36},b_{46},0,0,0)^{T}=2(c_{17},c_{27},c_{37},c_{47},0,0,0)^{T}={\cal{P}}^{(q)}_{0}\nabla_{q}h^{(q)}.$
Each of the Poisson matrices has 3 Casimirs and the $t_{h}$ and $t_{f}$ flows
have quadri-Hamiltonian representations, given below $\displaystyle{\bf
q}_{t_{h}}$ $\displaystyle=$
$\displaystyle{\cal{P}}_{0}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{4}\alpha_{0}\right)={\cal{P}}_{2}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{3}^{(q)}\nabla_{q}(2\alpha_{2}),$
$\displaystyle{\bf q}_{t_{f}}$ $\displaystyle=$
$\displaystyle{\cal{P}}_{0}^{(q)}\nabla_{q}f^{(q)}={\cal{P}}_{1}^{(q)}\nabla_{q}\left(-\frac{1}{8}h^{(q)}\right)={\cal{P}}_{2}^{(q)}\nabla_{q}\left(\frac{1}{4}\alpha_{0}\right)={\cal{P}}_{3}^{(q)}\nabla_{q}\alpha_{1},$
$\displaystyle{\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{0}={\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{0}^{(q)}\nabla_{q}\alpha_{2}={\cal{P}}_{1}^{(q)}\nabla_{q}f^{(q)}={\cal{P}}_{1}^{(q)}\nabla_{q}\alpha_{1}={\cal{P}}_{1}^{(q)}\nabla_{q}\alpha_{2}$
$\displaystyle={\cal{P}}_{2}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{2}^{(q)}\nabla_{q}f^{(q)}={\cal{P}}_{2}^{(q)}\nabla_{q}\alpha_{2}={\cal{P}}_{3}^{(q)}\nabla_{q}h^{(q)}={\cal{P}}_{3}^{(q)}\nabla_{q}f^{(q)}={\cal{P}}_{3}^{(q)}\nabla_{q}\alpha_{0}=0.$

In the $\bf\hat{Q}$ space, we have

${\cal{P}}^{(\hat{Q})}_{0}=\frac{1}{\hat{\beta}_{0}}\begin{pmatrix}0&-4\hat{Q}_{1}&4\hat{P}_{2}\\!-\\!3\hat{Q}_{1}\hat{Q}_{2}&-\frac{3}{2}\hat{Q}_{2}\hat{P}_{1}\\!-\\!\frac{3}{2}\hat{Q}_{1}^{2}\\!+\\!\frac{3}{4}\hat{Q}_{2}^{3}\\!-\\!4\hat{\beta}_{2}&a_{15}&0&0\\\
4\hat{Q}_{1}&0&0&0&a_{25}&0&0\\\
3\hat{Q}_{1}\hat{Q}_{2}-4\hat{P}_{2}&0&0&\frac{3}{2}\hat{\beta}_{0}&a_{35}&0&0\\\
\frac{3}{2}\hat{Q}_{2}\hat{P}_{1}\\!+\\!\frac{3}{2}\hat{Q}_{1}^{2}\\!-\\!\frac{3}{4}\hat{Q}_{2}^{3}\\!+\\!4\hat{\beta}_{2}&0&-\frac{3}{2}\hat{\beta}_{0}&0&a_{45}&0&0\\\
-a_{15}&-a_{25}&-a_{35}&-a_{45}&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{pmatrix},$ (53a)
${\cal{P}}^{(\hat{Q})}_{2}=\begin{pmatrix}0&\frac{4}{3}&0&\hat{Q}_{1}&0&b_{16}&0\\\
-\frac{4}{3}&0&0&-\hat{Q}_{2}&0&b_{26}&0\\\ 0&0&0&-\hat{P}_{1}&0&b_{36}&0\\\
-\hat{Q}_{1}&\hat{Q}_{2}&\hat{P}_{1}&0&0&b_{46}&0\\\ 0&0&0&0&0&0&0\\\
-b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0&0\\\ 0&0&0&0&0&0&0\end{pmatrix},$ (53b)
${\cal{P}}^{(\hat{Q})}_{3}=\begin{pmatrix}0&-4\hat{Q}_{2}&-4\hat{P}_{1}&-3\hat{Q}_{1}\hat{Q}_{2}&0&c_{16}&c_{17}\\\
4\hat{Q}_{2}&0&0&3\hat{Q}_{2}^{2}-4\hat{P}_{1}&0&c_{26}&c_{27}\\\
4\hat{P}_{1}&0&0&3\hat{Q}_{2}\hat{P}_{1}&0&c_{36}&c_{37}\\\
3\hat{Q}_{1}\hat{Q}_{2}&4\hat{P}_{1}-3\hat{Q}_{2}^{2}&-3\hat{Q}_{2}\hat{P}_{1}&0&0&c_{46}&c_{47}\\\
0&0&0&0&0&0&0\\\ -c_{16}&-c_{26}&-c_{36}&-c_{46}&0&0&0\\\
-c_{17}&-c_{27}&-c_{37}&-c_{47}&0&0&0\end{pmatrix},$ (53c) where
$\displaystyle(a_{15},a_{25},a_{35},a_{45},0,0,0)^{T}=(c_{16},c_{26},c_{36},c_{46},0,0,0)^{T}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}f^{(\hat{Q})},$
$\displaystyle(b_{16},b_{26},b_{36},b_{46},0,0,0)^{T}=\frac{2}{3}(c_{17},c_{27},c_{37},c_{47},0,0,0)^{T}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}.$

Each of the Poisson matrices has 3 Casimirs and the $t_{h}$ and $t_{f}$ flows
have quadri-Hamiltonian representations as below

$\displaystyle{\bf\hat{Q}}_{t_{h}}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\left(\frac{1}{6}f^{(\hat{Q})}\right)={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\left(\frac{2}{3}\hat{\beta}_{2}\right),$
(53d)
$\displaystyle{\bf\hat{Q}}_{t_{f}}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}f^{(\hat{Q})}={\cal{P}}_{0}^{(\hat{Q})}\left(\frac{1}{2}\hat{\beta}_{0}^{2}\right)={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}(3h^{(\hat{Q})})={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1},$
$\displaystyle{\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{1}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{1}={\cal{P}}_{0}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}$
$\displaystyle={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}f^{(\hat{Q})}={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}={\cal{P}}_{2}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{2}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}h^{(\hat{Q})}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}f^{(\hat{Q})}={\cal{P}}_{3}^{(\hat{Q})}\nabla_{\hat{Q}}\hat{\beta}_{0}=0.$
(53e)

###  6.4 The Quadri-Hamiltonian Form of (28)

The Poisson tensors (52) of (44) (=(21c)) were derived by using the Miura maps
of Section 6.2. However, these Poisson tensors are an intrinsic property of
(21c), independent of our method of construction, so can be understood in the
context of Section 4. We can therefore use the Poisson map (29) to construct
an additional pair of Poisson brackets for (28):

$\displaystyle{\cal{P}}^{(Q)}_{2}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&0&2{Q}_{1}&{Q}_{2}&0&a_{16}&0\\\
0&0&{Q}_{2}&0&0&a_{26}&0\\\ -2{Q}_{1}&-{Q}_{2}&0&{P}_{2}&0&a_{36}&0\\\
-{Q}_{2}&0&-{P}_{2}&0&0&a_{46}&0\\\ 0&0&0&0&0&0&0\\\
-a_{16}&-a_{26}&-a_{36}&-a_{46}&0&0&0\\\ 0&0&0&0&0&0&0\end{pmatrix},$
$\displaystyle{\cal{P}}^{(Q)}_{3}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&0&4{Q}_{1}^{2}+{Q}_{2}^{2}&2{Q}_{1}{Q}_{2}&0&b_{16}&b_{17}\\\
0&0&2{Q}_{1}{Q}_{2}&{Q}_{2}^{2}&0&b_{26}&b_{27}\\\
-4{Q}_{1}^{2}-{Q}_{2}^{2}&-2{Q}_{1}{Q}_{2}&0&2{Q}_{1}P_{2}&0&b_{36}&b_{37}\\\
-2{Q}_{1}{Q}_{2}&-{Q}_{2}^{2}&-2{Q}_{1}{P}_{2}&0&0&b_{46}&b_{47}\\\
0&0&0&0&0&0&0\\\ -b_{16}&-b_{26}&-b_{36}&-b_{46}&0&0&0\\\
-b_{17}&-b_{27}&-b_{37}&-b_{47}&0&0&0\end{pmatrix},$ where
$(a_{16},a_{26},a_{36},a_{46},0,0,0)^{T}=-(b_{17},b_{27},b_{37},b_{47},0,0,0)^{T}=2{\cal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}$
and
$(b_{16},b_{26},b_{36},b_{46},0,0,0)^{T}=2{\cal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}$,
where $h^{(Q)}$ and $f^{(Q)}$ are given respectively by (28) and (26e), for
$N=2$.

Each of these has 3 Casimirs:

${\cal{P}}_{2}^{(Q)}\nabla_{Q}f^{(Q)}={\cal{P}}_{2}^{(Q)}\nabla_{Q}\beta_{2}={\cal{P}}_{2}^{(Q)}\nabla_{Q}b_{1}={\cal{P}}_{3}^{(Q)}\nabla_{Q}f^{(Q)}={\cal{P}}_{3}^{(Q)}\nabla_{Q}\beta_{2}={\cal{P}}_{3}^{(Q)}\nabla_{Q}h^{(Q)}.$
(54b)

Taken together with ${\cal{P}}^{(Q)}_{0}$ and ${\cal{P}}^{(Q)}_{1}$ of (30a),
this renders the flows of $h^{(Q)}$ and $f^{(Q)}$ as quadri-Hamiltonian :

$\displaystyle{\bf Q}_{t_{h}}$ $\displaystyle=$
$\displaystyle{\cal{P}}_{1}^{(Q)}\nabla_{Q}h^{(Q)}={\cal{P}}_{0}^{(Q)}\nabla_{Q}f^{(Q)}={\cal{P}}_{2}^{(Q)}\nabla_{Q}\left(\frac{1}{2}b_{0}\right)={\cal{P}}_{3}^{(Q)}\nabla_{Q}\left(-\frac{1}{2}b_{1}\right),$
$\displaystyle{\bf Q}_{t_{f}}$ $\displaystyle=$
$\displaystyle{\cal{P}}_{1}^{(Q)}\nabla_{Q}f^{(Q)}={\cal{P}}_{0}^{(Q)}\nabla_{Q}(-\beta_{2})={\cal{P}}_{2}^{(Q)}\nabla_{Q}h^{(Q)}={\cal{P}}_{3}^{(Q)}\nabla_{Q}\left(\frac{1}{2}b_{0}\right).$

####  6.4.1 Recursion Operators

For the PDE flows, we build a recursion operator ${\cal R}=B_{1}B_{0}^{-1}$,
which includes the formal inverse $\partial_{x}^{-1}$ of the differential
operator $\partial_{x}$. Then $B_{n+1}={\cal R}B_{n}$, which can be formally
continued ad infinitum, but only a finite number of the resulting operators
are local, meaning that they depend upon only differential operators. For the
$M$ component cKdV systems, only the first $M+1$ operators are locally
defined.

For the stationary flows the situation is more complicated, since each Poisson
tensor ${\cal P}$ has only rank 4, so cannot be inverted. However, since the
Poisson map (29) can be restricted to the 4 components
$\phi:(Q_{i},P_{i})\rightarrow(q_{i},p_{i})$ (and the restriction is
invertible), then the corresponding $4\times 4$ submatrix of
${\cal{P}}^{(Q)}_{0}$, which we call ${\cal Q}_{0}^{(Q)}$, is invertible.
Defining ${\cal Q}_{1}^{(Q)}$ to be the corresponding $4\times 4$ submatrix of
${\cal P}_{1}^{(Q)}$, we can define a “recursion operator”:

${\cal R}^{(Q)}={\cal Q}_{1}^{(Q)}\left({\cal
Q}_{0}^{(Q)}\right)^{-1}=\left(\begin{array}[]{cccc}2Q_{1}&Q_{2}&0&0\\\
Q_{2}&0&0&0\\\ 0&P_{2}&2Q_{1}&Q_{2}\\\ -P_{2}&0&Q_{2}&0\\\
\end{array}\right).$ (55a) Defining ${\cal Q}_{n+1}^{(Q)}={\cal R}^{(Q)}{\cal
Q}_{n}^{(Q)}$, we find ${\cal
Q}_{2}^{(Q)}=\left(\begin{array}[]{cccc}0&0&2Q_{1}&Q_{2}\\\ 0&0&Q_{2}&0\\\
-2Q_{1}&-Q_{2}&0&P_{2}\\\ -Q_{2}&0&-P_{2}&0\\\ \end{array}\right),\;\;\;{\cal
Q}_{3}^{(Q)}=\left(\begin{array}[]{cccc}0&0&4Q_{1}^{2}+Q_{2}^{2}&2Q_{1}Q_{2}\\\
0&0&2Q_{1}Q_{2}&Q_{2}^{2}\\\
-4Q_{1}^{2}-Q_{2}^{2}&-2Q_{1}Q_{2}&0&2Q_{1}P_{2}\\\
-2Q_{1}Q_{2}&-Q_{2}^{2}&-2Q_{1}P_{2}&0\\\ \end{array}\right),$ (55b) which
should be compared with the formulae of (LABEL:t2-P23Q).

######  Remark 6.1 (In the ${\bf q}-$space)

We can similarly define ${\cal R}^{(q)}={\cal Q}_{1}^{(q)}\left({\cal
Q}_{0}^{(q)}\right)^{-1}$, which also generates ${\cal Q}_{n}^{(q)}$, for
higher Poisson brackets. In the usual way, these recursion operators are
intertwined by the Jacobian $J_{\phi}$ of the map $\phi$, with ${\cal
R}^{(q)}J_{\phi}=J_{\phi}{\cal R}^{(Q)}$.

##### Universality of these formulae:

The Poisson map (29) has a universal character. The first 4 components (our
map $\phi$) are fixed, but $\alpha_{0}$ and $\beta_{2}$ must be nonzero, in
order to guarantee that $h^{(Q)}$ and $f^{(q)}$ are nontrivial. This minimal
requirement is satisfied by the hierarchies for all values of $M$ in (2a),
with the minimal achieved by $M=1$ (the KdV case). As we increase $M$, then we
add more parameters, with the corresponding relations, as seen in the case of
$M=2$ and $M=3$. We conjecture that the formula for ${\cal Q}_{n}^{(Q)}$
represents this part of ${\cal P}_{n}^{(Q)}$, for all $n$.

The $M$ component cKdV hierarchy has $M+1$ local Hamiltonian operators, but
the modifications have a diminishing number, as depicted in Figure 1. However,
each stationary hierarchy has $M+1$ local Poisson brackets $P_{k}^{(m)}$, as
described at the beginning of Section 5. Unlike the Poisson map (29), the
corresponding one between $\bf\hat{Q}$ and $\bf q$ cannot be restricted to the
first 4 components, since $\hat{\beta}_{0}$ enters the formulae. This leaves a
vestige of nonlocality in the formulae for $P_{k}^{(m)}$, with $k<m$, in that
$\hat{\beta}_{j}$ appear explicitly in the $4\times 4$ submatrices. The
corresponding recursion operators are given by ${\cal R}={\cal
Q}_{m+1}^{(m)}\left({\cal Q}_{m}^{(m)}\right)^{-1}$ and only connect ${\cal
Q}_{k}^{(m)}$, for $k\geq m$.

##  7 The Poisson Brackets of the General Parabolic Potentials

We see that the structure of the Poisson bracket $P_{0}^{(Q)}$ for the $t_{2}$
flow in the KdV and DWW cases (see [12]) and (30a) of this paper have
identical form (up to additional zero rows and columns), depending upon the
function $f^{(Q)}$. The Poisson bracket $P_{2}^{(Q)}$ is similarly common to
the DWW case and (LABEL:t2-P23Q) of this paper. In all these cases, the
Hamiltonian $h^{(Q)}$ belongs to the standard series known to be separable in
parabolic coordinates (see, again, Equation 2.2.41 in [16]).

Here we consider the general case of potential $U(Q_{1},Q_{2})$, separable in
parabolic coordinates. In all our examples (such as (28)) the potential
$U(Q_{1},Q_{2})$ depends upon some additional parameters in a specific way.
These parameters arose as Casimirs of the (degenerate) canonical bracket,
whilst their coefficients are mainly related to the Casimirs of the original
Poisson brackets of the corresponding PDE.

For our initial calculation of the general form of $U(Q_{1},Q_{2})$, we
suppress this parametric dependence, but when we consider additional Poisson
brackets these parameters explicitly appear as an essential ingredient.

###  7.1 Separation Coordinates

The existence of a pair of commuting integrals, which are quadratic in
momenta, means that we can simultaneously diagonalise the quadratic parts by
constructing separation variables. Generalising our $h^{(Q)}$ and $f^{(Q)}$,
consider

$h^{(Q)}=\frac{1}{2}(P_{1}^{2}+P_{2}^{2})+U(Q_{1},Q_{2}),\quad
f^{(Q)}=P_{2}\,(Q_{2}P_{1}-Q_{1}P_{2})+W(Q_{1},Q_{2}),$ (56a) where $U$ and
$W$ take specific forms in our examples, but here are considered arbitrary.
Using the canonical Poisson bracket $P_{1}^{(Q)}$, we find
$\\{f^{(Q)},h^{(Q)}\\}_{1}=0\quad\Rightarrow\quad
W_{Q_{1}}=Q_{2}U_{Q_{2}},\;\;\;W_{Q_{2}}=Q_{2}U_{Q_{1}}-2Q_{1}U_{Q_{2}},$
(56b) whose integrability condition implies a second order, linear hyperbolic
equation for $U$:
$Q_{2}\left(U_{Q_{1}Q_{1}}-U_{Q_{2}Q_{2}}\right)-2Q_{1}U_{Q_{1}Q_{2}}=3U_{Q_{2}}.$
(56c) The characteristic coordinates for this equation (equivalently, the
separation coordinates for the pair (56a)) are
$u=\sqrt{Q_{1}^{2}+Q_{2}^{2}}+Q_{1},\;\;v=\sqrt{Q_{1}^{2}+Q_{2}^{2}}-Q_{1}\;\;\;\Rightarrow\;\;\;\left((u+v)U\right)_{uv}=0,$
(56d) leading to $U=\frac{A_{1}(u)+A_{2}(v)}{u+v}\quad\mbox{and}\quad
W=\frac{vA_{1}(u)-uA_{2}(v)}{u+v}.$ (56e)

###  7.2 The Jacobi Identity for Non-Canonical Brackets

The non-canonical brackets have nontrivial entries in the rows and columns
corresponding to the Casimirs of the canonical bracket. The equations of the
Jacobi identity therefore include derivatives with respect to these
parameters. These can be solved to get explicit dependence of $U$ and $W$ on
the parameters, which are given the generic names $\kappa_{i}$ in this
section.

For these calculations, it is more convenient to use the general form with
$U(Q_{1},Q_{2},\kappa_{1},\kappa_{2},\kappa_{3})$ and
$W(Q_{1},Q_{2},\kappa_{1},\kappa_{2},\kappa_{3})$, depending upon parameters
$\kappa_{i}$, associated with the last 3 columns of the Poisson brackets
$P_{k}^{(Q)}$.

##### The Jacobi Identity for $P_{0}^{(Q)}$:

Since this Poisson matrix (see (30a)) only has additional entries in the
$\kappa_{1}$ column, the Jacobi identity just implies constraints on the
$\kappa_{1}$ dependence of both $U$ and $W$:

$W_{Q_{1}\kappa_{1}}+\frac{2}{Q_{2}^{2}}=W_{Q_{2}\kappa_{1}}-\frac{4Q_{1}}{Q_{2}^{3}}=0\quad\Rightarrow\quad\left\\{\begin{array}[]{l}W=-\frac{2\kappa_{1}Q_{1}}{Q_{2}^{2}}+W_{1}(Q_{1},Q_{2},\kappa_{2},\kappa_{3}),\\\\[4.2679pt]
U=\frac{\kappa_{1}}{Q_{2}^{2}}+U_{1}(Q_{1},Q_{2},\kappa_{2},\kappa_{3}),\end{array}\right.$
(57a) with the formula for $U$ being derived by using (56b).

##### The Jacobi Identity for $P_{2}^{(Q)}$:

Since this Poisson matrix (see (LABEL:t2-P23Q)) only has additional entries in
the $\kappa_{2}$ column, the Jacobi identity just implies constraints on the
$\kappa_{2}$ dependence of both $U$ and $W$:

$U_{Q_{1}\kappa_{2}}+1=U_{Q_{2}\kappa_{2}}=0\quad\Rightarrow\quad\left\\{\begin{array}[]{l}U_{1}=-\kappa_{2}Q_{1}+U_{2}(Q_{1},Q_{2},\kappa_{3}),\\\\[4.2679pt]
W_{1}=-\frac{1}{2}\kappa_{2}Q_{2}^{2}+W_{2}(Q_{1},Q_{2},\kappa_{3}),\end{array}\right.$
(57b)

with the formula for $W$ being derived by using (56b).

##### The Jacobi Identity for $P_{3}^{(Q)}$:

Since this Poisson matrix (see (LABEL:t2-P23Q)) has additional entries in both
the $\kappa_{2}$ and $\kappa_{3}$ columns, the Jacobi identity implies
constraints on both the $\kappa_{2}$ and $\kappa_{3}$ dependence of both $U$
and $W$, but the $\kappa_{2}$ equations are identically satisfied. We are left
with some equations:

$\left.\begin{array}[]{l}U_{Q_{1}\kappa_{3}}-4Q_{1}=U_{Q_{2}\kappa_{3}}-Q_{2}=0\\\\[4.2679pt]
W_{Q_{1}\kappa_{3}}-Q_{2}^{2}=W_{Q_{2}\kappa_{3}}-2Q_{1}Q_{2}=0\end{array}\right\\}\quad\Rightarrow\quad\left\\{\begin{array}[]{l}U_{2}=\frac{1}{2}\kappa_{3}(4Q_{1}^{2}+Q_{2}^{2})+U_{3}(Q_{1},Q_{2}),\\\\[4.2679pt]
W_{2}=\kappa_{3}Q_{1}Q_{2}^{2}+W_{3}(Q_{1},Q_{2}),\end{array}\right.$ (57c)

giving

$\displaystyle h^{(Q)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}^{2}+P_{2}^{2})+\frac{A_{1}(u)+A_{2}(v)}{2\sqrt{Q_{1}^{2}+Q_{2}^{2}}}+\frac{\kappa_{1}}{Q_{2}^{2}}-\kappa_{2}Q_{1}+\frac{1}{2}\kappa_{3}(4Q_{1}^{2}+Q_{2}^{2}),$
$\displaystyle f^{(Q)}$ $\displaystyle=$ $\displaystyle
P_{2}\,(Q_{2}P_{1}-Q_{1}P_{2})+\frac{vA_{1}(u)-uA_{2}(v)}{2\sqrt{Q_{1}^{2}+Q_{2}^{2}}}-\frac{2\kappa_{1}Q_{1}}{Q_{2}^{2}}-\frac{1}{2}\kappa_{2}Q_{2}^{2}+\kappa_{3}Q_{1}Q_{2}^{2},$

where $u,\,v$ are given by (56d). We see that (28) and (26e) (with $N=2$) have
precisely this form, but with a specific choice of the functions $A_{i}$ and
$(\kappa_{1},\kappa_{2},\kappa_{3})=(\beta_{2},b_{0},b_{1})$.

###### Remark 7.1

Curiously, the same functional coefficients of $\kappa_{i}$ have arisen in two
seemingly different ways: here they were a consequence of the Jacobi identity,
whilst previously they arose as Casimirs of the Hamiltonian structures in the
PDE context.

###### Remark 7.2

These $\kappa_{i}$ terms correspond to

$A_{1}^{\kappa}(u)=\frac{\kappa_{1}}{u}-\frac{1}{2}\kappa_{2}u^{2}+\frac{1}{2}\kappa_{3}u^{3},\quad
A_{2}^{\kappa}(v)=\frac{\kappa_{1}}{v}+\frac{1}{2}\kappa_{2}v^{2}+\frac{1}{2}\kappa_{3}v^{3}.$

In summary, we have

###### Proposition 7.3

The general system (LABEL:genParab-hQfQ-solk123) is quadri-Hamiltonian, with
Poisson brackets $P_{k}^{(Q)}$, for $k=0,\dots,3$. The reduction with
$\kappa_{3}=0$ is tri-Hamiltonian, whilst that with $\kappa_{2}=\kappa_{3}=0$
is bi-Hamiltonian.

###  7.3 The Poisson Map Between $\bf q$ and $\bf Q$ Variables

The Poisson map (29) (with a relabelling of parameters), can be used in this
more general context. The formulae are dictated purely by the “kinetic” parts
of the various functions, not the form of the potential. We then find

$\displaystyle h^{(q)}$ $\displaystyle=$ $\displaystyle
5q_{2}p_{1}^{2}-4p_{1}p_{2}+\frac{\bar{u}A_{1}(\bar{u})-\bar{v}A_{2}(\bar{v})}{16(\bar{u}+\bar{v})}+\alpha_{0}q_{2}+\alpha_{1}(4q_{1}+q_{2}^{2})-4\alpha_{2}q_{2}(2q_{1}+q_{2}^{2}),$
$\displaystyle f^{(q)}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(4q_{1}-5q_{2}^{2}\right)p_{1}^{2}+4q_{2}p_{1}p_{2}-2p_{2}^{2}-\frac{\bar{u}\bar{v}(A_{1}(\bar{u})+A_{2}(\bar{v}))}{64(\bar{u}+\bar{v})}$
$\displaystyle\hskip
85.35826pt+\frac{1}{4}\alpha_{0}\left(4q_{1}+3q_{2}^{2}\right)-\frac{1}{2}\alpha_{1}q_{2}\left(4q_{1}+3q_{2}^{2}\right)-\frac{1}{4}\alpha_{2}(4q_{1}+q_{2}^{2})(4q_{1}+3q_{2}^{2}),$
where
$\bar{u}=\sqrt{-8q_{1}-5q_{2}^{2}}+q_{2},\quad\bar{v}=\sqrt{-8q_{1}-5q_{2}^{2}}-q_{2},\quad\kappa_{2}=16\alpha_{1},\quad\kappa_{3}=8\alpha_{2}.$
(58b)

Again, the coefficients of $\alpha_{i}$ are the ones required in order that
the Poisson brackets (52) satisfy the Jacobi identity. Again, $h^{(q)}$ and
$f^{(q)}$ commute with respect to each of these Poisson brackets.

###### Remark 7.4

The specific case of (21c) and (25b) corresponds to
$A_{1}=\frac{1}{128}\bar{u}^{6},\;A_{2}=-\frac{1}{128}\bar{v}^{6}$.

###  7.4 Compatibility with the Lax Representation

We now consider the constraints on $h^{(Q)}$ and $f^{(Q)}$ imposed by the
existence of the Lax matrix (26a), whose $a_{21}$ component is given by (26b),
but with $h^{(Q)}$ and $f^{(Q)}$ given by (56a). The term $\frac{\partial
f^{(Q)}}{\partial Q_{1}}$ includes $W_{Q_{1}}$, but this can be written as
$Q_{2}U_{Q_{2}}$, by using (56b). The characteristic equation of the resulting
$L^{(2)}$ is of the form

$z^{2}+\sum_{i=0}^{3}\chi_{i}\lambda^{i}+256\kappa_{3}\lambda^{4}-256\lambda^{7}=0,$
(59a) where each coefficient must be a constant (of motion). In particular,
$\chi_{0}=-Q_{2}^{3}U_{Q_{2}}+\frac{1}{16}Q_{1}Q_{2}^{4}(8Q_{1}^{2}+3Q_{2}^{2})+\kappa_{3}Q_{2}^{4},\quad\chi_{3}=80Q_{1}^{4}+48Q_{1}^{2}Q_{2}^{2}+3Q_{2}^{4}+256\kappa_{3}Q_{1}-64U_{Q_{1}},$
(59b) so setting $\chi_{0}=2\kappa_{1}$ and $\chi_{3}=64\kappa_{2}$ gives
equations for $U_{Q_{i}}$, which are compatible and have solution
$U(Q_{1},Q_{2})=\frac{\kappa_{1}}{Q_{2}^{2}}-\kappa_{2}Q_{1}+\frac{1}{2}\kappa_{3}(4Q_{1}^{2}+Q_{2}^{2})+\frac{1}{64}Q_{1}(4Q_{1}^{2}+Q_{2}^{2})(4Q_{1}^{2}+3Q_{2}^{2}),$
(59c) reproducing the form of $h^{(Q)}$ in (28) (with
$(\kappa_{1},\kappa_{2},\kappa_{3})=(\beta_{2},b_{0},b_{1})$). With these
formulae, the characteristic equation (59a) reduces to
$z^{2}+2\kappa_{1}+8f^{(Q)}\lambda-32h^{(Q)}\lambda^{2}+64\kappa_{2}\lambda^{3}+256\kappa_{3}\lambda^{4}-256\lambda^{7}=0,$
(59d) which is just (26c), with $N=2$.

###### Remark 7.5

We can easily add lower degree terms, corresponding to the stationary KdV and
DWW flows. Clearly we cannot expect to represent the general case of
(LABEL:genParab-hQfQ-solk123) in terms of this class of Lax matrix.

###  7.5 The Sequence of Polynomial Potentials

Whilst the Poisson brackets allow any of the potentials in (LABEL:genParab-
hQfQ-solk123), we have seen that the Lax matrix (26a) fixes the potential to
be (59c). The homogeneous polynomial forms of (56e), of degree $n$, are given
by the choice $A_{1}(u)=u^{n+1},\,A_{2}(v)=(-1)^{n}v^{n+1}$ and it is these
particular potentials which appear in stationary reductions of the $M$
component cKdV hierarchy, corresponding to $M=n-2$.

In the Lax matrix (26a), the only element which depends upon $M$ is the
$a_{21}$ component (26b), which can be written

$a_{21}^{(M)}=-u^{(M)}\,\left(16\lambda^{2}+8Q_{1}\lambda-
Q_{2}^{2}\right)+4\frac{\partial h^{(Q)}}{\partial
Q_{1}}\lambda-\frac{\partial f^{(Q)}}{\partial
Q_{1}},\;\;\mbox{where}\;\;u^{(M)}=\sum_{i=0}^{M-1}u_{i}\lambda^{i}-\lambda^{M},$
(60a) with each $u_{i}$ replaced by an appropriate function of $Q_{1},Q_{2}$.
Going through the stationary reduction route to obtain these formulae for
$u_{i}$ is laborious. However, it is possible to divorce the construction of
$L^{(2)}$ from the stationary reduction process and recursively build
$a_{21}^{(M)}$. In fact, we can recursively build $u^{(M)}$, with the formula
$u^{(M+1)}=\lambda\,u^{(M)}+g_{M+1}(Q_{1},Q_{2}),$ (60b) where $u^{(M)}$ is
known, and a single unknown function $g_{M+1}(Q_{1},Q_{2})$ is to be
determined, along with the new $h^{(Q)}$ and $f^{(Q)}$. We follow the approach
of Section 7.4, using the characteristic equation of $L^{(2)}$, with
$u^{(M+1)}$. To illustrate the procedure, consider the Lax matrix (26a), with
$u^{(3)}=\frac{1}{16}Q_{1}(8Q_{1}^{2}+3Q_{2}^{2})+\kappa_{3}-\frac{1}{8}(6Q_{1}^{2}+Q_{2}^{2})\lambda+Q_{1}\lambda^{2}-\lambda^{3},\;\;\;\mbox{and}\;\;u^{(4)}=\lambda\,u^{(3)}+g_{4}(Q_{1},Q_{2}).$
(60c) The characteristic equation of the resulting $L^{(2)}$, is
$z^{2}+\sum_{i=0}^{4}\chi_{i}\lambda^{i}+256\kappa_{3}\lambda^{5}-256\lambda^{8}=0,$
(60d) with $\displaystyle\chi_{4}$ $\displaystyle=$ $\displaystyle
80Q_{1}^{4}+48Q_{1}^{2}Q_{2}^{2}+3Q_{2}^{4}+256\kappa_{3}Q_{1}+256g_{4},$
$\displaystyle\chi_{3}$ $\displaystyle=$
$\displaystyle-64U_{Q_{1}}+Q_{1}(32Q_{1}^{4}+8Q_{1}^{2}Q_{2}^{2}-3Q_{2}^{4})+32\kappa_{3}(2Q_{1}^{2}-Q_{2}^{2})+256Q_{1}g_{4},$
(60e) $\displaystyle\chi_{0}$ $\displaystyle=$ $\displaystyle
Q_{2}^{3}\left(Q_{2}g_{4}-U_{Q_{2}}\right).$ Setting
$\chi_{4}=256\kappa_{4},\,\chi_{3}=64\kappa_{2},\,\chi_{0}=2\kappa_{1}$, we
find $\displaystyle g_{4}(Q_{1},Q_{2})$ $\displaystyle=$
$\displaystyle\kappa_{4}-\kappa_{3}Q_{1}-\frac{1}{256}\left(80Q_{1}^{4}+48Q_{1}^{2}Q_{2}^{2}+3Q_{2}^{4}\right),$
(60f) $\displaystyle U(Q_{1},Q_{2})$ $\displaystyle=$
$\displaystyle\frac{\kappa_{1}}{Q_{2}^{2}}-\kappa_{2}Q_{1}+\frac{1}{2}\kappa_{4}(4Q_{1}^{2}+Q_{2}^{2})-\frac{1}{2}\kappa_{3}Q_{1}(2Q_{1}^{2}+Q_{2}^{2})$
(60g) $\displaystyle\hskip
128.0374pt-\frac{1}{512}\left(64Q_{1}^{6}+80Q_{1}^{4}Q_{2}^{2}+24Q_{1}^{2}Q_{2}^{4}+Q_{2}^{6}\right).$

###### Remark 7.6

The Lax equations for $L^{(2)}$, with $a_{21}^{(M)}$ are just (26f), with

$L^{(1)}=\left(\begin{array}[]{cc}P_{1}&-2(2\lambda+Q_{1})\\\
b_{21}&-P_{1}\end{array}\right),\quad U=\left(\begin{array}[]{cc}0&1\\\
-u^{(M)}&0\end{array}\right),$

where $b_{21}=2(2\lambda+Q_{1})u^{(M)}-\frac{\partial h^{(Q)}}{\partial
Q_{1}}$.

##  8 Conclusions

In this paper we have considered the general structure of the $t_{1}$ and
$t_{2}$ stationary flows of the $M$ component cKdV hierarchy, giving detailed
formulae for the case $M=3$. One of our reductions, using a squared
eigenfunction representation, gives an $N$ component, superintegrable system
with a Lax pair. For $N=2$, we presented Poisson maps which led to multi-
Hamiltonian representations.

The $t_{2}$ stationary flow is separable in parabolic coordinates and the
Poisson brackets were generalised to include the entire class of separable
potentials. The Jacobi identity imposed a specific dependence on the
parameters $\kappa_{i}$. The recursive procedure of Section 7.5 allows us to
build the Lax matrices without any reference to the stationary reduction.
Similarly, the “recursion operator” of Section 6.4.1 allows us to build higher
Poisson tensors, without needing to go through the entire stationary reduction
process.

An important open problem is to generalise these Poisson brackets to the case
$N\geq 3$. In particular, for $N=3$ it would be interesting to find multi-
Hamiltonian formulations for the generalisations given in Section 6 of [12],
which are related to generalised parabolic coordinates and also the rational
Calogero-Moser model. This connection first arose in [11], in which we also
presented a conformally flat extension. Neither a Lax representation nor a
multi-Poisson formulation are known for this case.

The matrices of (2a) have a polynomial dependence on $\lambda$. In [1, 2] we
gave a more general formulation, with rational dependence on $\lambda$, which
includes the Ito hierarchy. Furthermore, the latter includes both positive and
negative hierarchies. It would be interesting to know whether further cases of
the general parabolic potentials of Section 7 can be given a Lax
representation in this way.

Stationary flows are related to the $t$ translation symmetry of the
corresponding PDE. Using the scaling symmetry, it is possible to reduce to
Painlevé type equations (see [13, 14]). Such reductions would be interesting
to investigate.

### Acknowledgements

This work was supported by the National Natural Science Foundation of China
(grant no. 11871396).

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