Integrability, intertwiners and non-linear algebras
in Calogero models
Francisca Carrillo–Moralesa, Francisco Correaa and Olaf Lechtenfeldb

aInstituto de Ciencias Físicas y Matemáticas
Universidad Austral de Chile, Casilla 567, Valdivia, Chile
bInstitut für Theoretische Physik and Riemann Center for Geometry and Physics
Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany

###### Abstract

For the rational quantum Calogero systems of type $A_{1}{\oplus}A_{2}$,
$AD_{3}$ and $BC_{3}$, we explicitly present complete sets of independent
conserved charges and their nonlinear algebras. Using intertwining (or shift)
operators, we include the extra ‘odd’ charges appearing for integral
couplings. Formulæ for the energy eigenstates are used to tabulate the low-
level wave functions.

##  1 Introduction and summary

There is a vast literature on quantum Calogero (or Calogero–Moser–Sutherland)
models as a paradigm for a superintegrable system with a finite number of
degrees of freedom [1, 2]. Since these systems admit an analytic computation
of basically every detail, a variety of rich mathematical structures has been
uncovered (for reviews, see e.g. [3, 4]; for a recent generalization, see
[5]). Nevertheless, some of the latter have not been displayed very explicitly
or are hidden in mathematical literature difficult to penetrate for most
physicists.

Among these aspects is the nonlinear algebra formed by the $2N{-}1$ (or, for
integral coupling, $2N$) independent conserved charges in the case of the
rational rank-$N$ Calogero model [6, 7]. A related feature is the form of the
intertwiners (or shift operators). These relate the simultaneous eigenstates
of the $N$ Liouville charges at integer-spaced coupling values, thereby
providing an alternative access to those states for integral couplings and
allowing for the $2N$th extra charge $Q$.

For this reason, we provide completely explicit formulæ for the rank-3
rational quantum Calogero models, based on the Coxeter reflection groups
$A_{1}{\oplus}A_{2}$, $AD_{3}$ and $BC_{3}$ (but not $H_{3}$), for the
nonlinear algebras, the intertwiners and the energy eigenstates. In contrast
to the customary $A_{1}{\oplus}A_{2}$ model (describing three nonrelativistic
unit-mass particles on the infinite line and interacting pairwise via an
inverse-square two-body potential), the $AD_{3}$ and $BC_{3}$ models have been
investigated much less. Yet, even the $A_{1}{\oplus}A_{2}$ case has not been
fully analyzed: The center-of-mass sector (associated to the $A_{1}$ part) is
usually free, as one imposes translational invariance as a physical prejudice.
However, nothing prevents one from giving it its own (external) inverse-square
potential. In fact, such is completely natural in the bigger scope, and here
we present all details for this generalized three-particle system as well.

The organization of the paper is as follows. In the remainder of this
introduction, we introduce the Dunkl operators [8, 9] as our major tool for
the construction of conserved charges, then review how the conformal algebra
can be used to extend the set of Liouville charges to an infinite (but
functionally dependent) set of higher integrals of motion, and finally present
the concept of intertwining operators and how they give rise to additional
conserved charges in the special situation of integral coupling(s). Sections
2, 3 and 4 then treat the cases of $A_{1}{\oplus}A_{2}$, $AD_{3}$ and $BC_{3}$
in turn, showing in each instance a full set of independent conserved charges,
a formula for the energy eigenstates, the concrete form of the basic
intertwiners, and finally the nonlinear algebra of all charges including the
extra ones at integral coupling. Some lengthy expressions and lists of low-
lying energy eigenstates are delegated to an appendix.

###  1.1 Liouville charges

Let us consider a set $R^{+}$ of positive roots $\alpha$ for a Coxeter group
$W$ of reflections $s_{\alpha}$ in $\mathbb{R}^{N}\ni x$. Then, the Dunkl
operators associated to the standard Cartesian basis $\\{e_{i}\\}$ with
$i=1,\ldots,N$ are given by

$\mathcal{D}_{i}\ =\ \partial_{i}-\sum_{\alpha\in
R^{+}}\dfrac{g_{\alpha}\alpha_{i}}{(\alpha,x)}s_{\alpha}\qquad\text{with}\quad(\alpha,x)=\alpha(x){\quad{\rm
and}\quad}\alpha_{i}=(\alpha,e_{i})=\alpha(e_{i})\ ,$ (1)

where $\partial_{i}=\partial/\partial x^{i}$ for $x=e_{i}x^{i}$, and we
canonically identify $\mathbb{R}^{N}$ with its dual. The real coupling
constants $g_{\alpha}$ depend only on the Weyl orbit of $\alpha$, so we shall
encounter at most two values, either $g_{s}$ for the short roots and
$g_{\ell}$ for the long roots, or a single coupling $g$ in the simply-laced
case. A key role is played by the Weyl-invariant polynomials $\sigma_{k}(x)$
of degree $k$, because the restriction “${\mathrm{res}}$” of
$\sigma_{k}(\mathcal{D})$ for $\mathcal{D}\equiv\\{\mathcal{D}_{i}\\}$ to
Weyl-invariant functions yields constants of motion $I_{k}$ known as Liouville
integrals, for any (generalized) Calogero model. Since the Dunkl operators
mutually commute [8, 9], the Liouville integrals also commute with one
another. The $N$ lowest-order such polynomials will provide $N$ functionally
independent Liouville charges.

A universal Weyl-invariant polynomial is $\sigma_{2}(x)=(x,x)=:r^{2}$. The
corresponding integral $I_{2}$ is (minus twice) the Hamiltonian of the system,

$H\ =\ -\tfrac{1}{2}{\mathrm{res}}(\mathcal{D},\mathcal{D})\ =\
\tfrac{1}{2}\sum_{i=1}^{N}p_{i}^{2}+\sum_{\alpha\in
R^{+}}\frac{g_{\alpha}(g_{\alpha}{-}1)(\alpha,\alpha)}{2\,(\alpha,x)^{2}}\qquad\text{with}\quad
p_{i}=-{\mathrm{i}}\partial_{i}\ ,$ (2)

and is invariant under $g_{\alpha}\to 1{-}g_{\alpha}$ for any $\alpha$. For
$g_{\alpha}=0$ or $g_{\alpha}=1$, the (free) Liouville charges are simply
given by

$I_{k}\ =\ \sigma_{k}(p)\ .$ (3)

###  1.2 Energy spectrum

The energy spectrum is continuous with $E\geq 0$ but highly degenerate. The
$H$ eigenstates may be labelled by the energy $E$ and by $N{-}1$ additional
quantum numbers $\ell_{k}$ corresponding to the Weyl-invariant polynomials
$\sigma_{k}$ (other than $\sigma_{2}$) and combining in the generalized
angular momentum

$q\ =\ \sum_{\alpha}g_{\alpha}+{\sum_{\\{k\\}}}^{\prime}k\,\ell_{k}$ (4)

where the prime indicates leaving out $k{=}2$. They are conveniently found by
separating the Schrödinger equation in spherical coordinates
$(r,\vec{\theta})$, yielding a basis of eigenfunctions

$\Psi^{(\\{g_{\alpha}\\})}_{{\scriptscriptstyle E},\\{\ell_{k}\\}}(x)\ =\
r^{-(N-2)/2}\,J_{q+(N-2)/2}({\scriptstyle\sqrt{2E}}\,r)\
v^{(\\{g_{\alpha}\\})}_{\\{\ell_{k}\\}}(\vec{\theta})$ (5)

with a Bessel-type radial dependence. The angular part of the wave function is
best expressed as

$v^{(\\{g_{\alpha}\\})}_{\\{\ell_{k}\\}}(\vec{\theta})\ =\
r^{-q}{\Delta\vphantom{\big{|}}}^{g}\,h^{(\\{g_{\alpha}\\})}_{\\{\ell_{k}\\}}(x)\qquad\text{with}\quad{\Delta\vphantom{\big{|}}}^{g}\
=\ \prod_{\alpha}(\alpha,x)^{g_{\alpha}}$ (6)

and a Dunkl-deformed harmonic polynomial 111 It is annihilated by
$\tilde{H}={\Delta\vphantom{\big{|}}}^{-g}H\,{\Delta\vphantom{\big{|}}}^{g}=-\tfrac{1}{2}\sum_{i}\partial_{i}^{2}+O(g)$.

$h^{(\\{g_{\alpha}\\})}_{\\{\ell_{k}\\}}(x)\ =\
r^{N-2+2q}\,\Bigl{\\{}{\prod_{\\{k\\}}}^{\prime}\sigma_{k}(\mathcal{D})^{\ell_{k}}\Bigr{\\}}\,r^{-N+2-2\sum_{\alpha}g_{\alpha}}$
(7)

of degree $\sum^{\prime}_{\\{k\\}}k\,\ell_{k}$. It may be noted that these
states generally are not eigenstates of the other Liouville charges but can be
linearly combined to jointly diagonalize all of them.

###  1.3 Conformal algebra

The Hamiltonian (2) and the two operators

$D=\tfrac{1}{2}\sum_{i=1}^{N}(x^{i}p_{i}+p_{i}x^{i}){\qquad{\rm
and}\qquad}K=\tfrac{1}{2}\sum_{i=1}^{N}(x^{i})^{2}={{\textstyle\frac{1}{2}}}(x,x)$
(8)

form the basis of a conformal algebra $sl(2,\mathbb{R})$,

$[D,H]=2{\mathrm{i}}H\ ,\qquad[D,K]=-2{\mathrm{i}}K\
,\qquad[K,H]={\mathrm{i}}D\ ,$ (9)

where $D$ generates the scale transformations (or dilatations) and $K$ the
special conformal transformations.

This dynamical symmetry is enhanced by extending $H$ to the entire set
$\\{I_{k}\\}$ of Liouville charges, which produces an infinite quadratic
algebra [6, 7]. The dilatation operator yields a $\mathbb{Z}$ grading, but
each commutation with $K$ provides a new layer of operators, starting with

$\tfrac{1}{{\mathrm{i}}}[K,I_{k}]=kJ_{k}$ (10)

satisfying the commutation relations

$\tfrac{1}{{\mathrm{i}}}[D,J_{k}]=(k{-}2)J_{k}{\qquad{\rm
and}\qquad}\tfrac{1}{{\mathrm{i}}}[H,J_{k}]=-I_{k}\ .$ (11)

Note that $J_{1}=\sum_{i}x^{i}$ and $J_{2}=D$. There are two obvious ways to
build further integrals of motion, which will not be in involution however.
The first one admits explicit time dependence,

$\bar{J}_{k}\ :=\
J_{k}-t\,I_{k}\qquad\Rightarrow\qquad{\textstyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}}\bar{J}_{k}\
=\ \partial_{t}\bar{J}_{k}+{\mathrm{i}}[H,\bar{J}_{k}]\ =\ 0\ .$ (12)

The second one forms the antisymmetric combinations

$L_{k,\ell}\ :=\
\tfrac{1}{2}(I_{k}J_{\ell}+J_{\ell}I_{k})-\tfrac{1}{2}(I_{\ell}J_{k}+J_{k}I_{\ell})\qquad\Rightarrow\qquad[H,L_{k,\ell}]\
=\
-\tfrac{{\mathrm{i}}}{2}(I_{k}I_{\ell}+I_{\ell}I_{k}-I_{\ell}I_{k}-I_{k}I_{\ell})\
=\ 0\ .$ (13)

The $\bar{J}_{k}$ or the $L_{k,\ell}$ form overcomplete sets of constants of
motion, and there exist many options for a functionally independent complete
subset. Here, we choose

$\displaystyle F_{k}\ :=\ L_{2,k}\ =\
\\{H,J_{k}\\}-{{\textstyle\frac{1}{2}}}\\{I_{k},D\\}$ (14)

with the same $N$ lowest values for $k$ determined by the first Weyl-invariant
polynomials $\sigma_{k}$. Since $F_{2}\equiv 0$ by definition, these provide
$N{-}1$ additional integrals of motion, revealing the complete
superintegrability of the rational quantum Calogero model. Our choice of extra
integrals bears a close relation to the Casimir element of the conformal
algebra,

$C\ =\ KH+HK-\tfrac{1}{2}D^{2}\ ,$ (15)

which generates the $F_{k}$ directly from the $I_{k}$,

$\tfrac{1}{{\mathrm{i}}}[C,I_{k}]\ =\ k\,F_{k}\ .$ (16)

The quadratic algebra spanned by the $I_{k}$ and $F_{k}$ has been presented in
[7].

###  1.4 Intertwining operators

The previous results were obtained for generic real couplings $g_{\alpha}$.
For integer coupling values, there appears one additional independent constant
of motion, due to the invariance of $H$ under $g\to 1{-}g$ and the existence
of intertwining (or shift) operators $M(g)$. The latter are constructed from
Weyl anti-invariant polynomials $\tau_{m}(x)$, again by replacing the
arguments $x^{i}$ with the Dunkl operators $\mathcal{D}_{i}$ and restricting
the result to Weyl-symmetric functions, as we did for the construction of the
Liouville integrals. The degree $m$ of those polynomials and the number of
independent ones depend on the root system under consideration. In this sense,
intertwining operators have been studied under this approach in angular and
trigonometric Calogero models [10, 11, 12].

Let us be more concrete for the simply-laced situation, $g_{\alpha}=g$. Any
intertwiner $M(g)$ establishes a relation between the Liouville integrals at
integrally shifted couplings,

$M(g)\,I_{k}(g)\ =\ I_{k}(g{+}1)\,M(g){\qquad{\rm
and}\qquad}M(1{-}g)\,I_{k}(g)\ =\ I_{k}(g{-}1)\,M(1{-}g)\ ,$ (17)

and hence transports eigenstates of $I_{k}(g)$ to eigenstates of
$I_{k}(g{+}1)$ (or zero). In particular,

$M(g)\,\Psi^{(g)}_{{\scriptscriptstyle E},\\{\ell_{k}\\}}(x)\ =\
\sum_{\\{k\\}}c_{\\{\ell_{k}\\}}^{\\{\ell^{\prime}_{k}\\}}(g)\,\Psi^{(g+1)}_{{\scriptscriptstyle
E},\\{\ell^{\prime}_{k}\\}}(x)\ ,$ (18)

with some coefficients $c_{\\{\ell_{k}\\}}^{\\{\ell^{\prime}_{k}\\}}(g)$. In
this way, simultaneous $I_{k}$ eigenstates at integer coupling can be obtained
from free eigenstates by a successive application of shift operators. As
another consequence, by shifting the coupling up and then down again, the
operator $M(-g)M(g)$ commutes with all Liouville charges,

$[M(-g)M(g),I_{k}(g)]\ =\ 0\ ,$ (19)

but it is not a new integral of motion since it is expressed in terms of them,

$M(-g)M(g)\ =\ \mathcal{R}(I(g))\qquad\textrm{for}\quad I=\\{I_{k}\\}\ .$ (20)

This polynomial of the $I_{k}$ must not depend on $g$ explicitly (take all
$(\alpha,x)\to\infty$). Therefore, one may easily compute it from the free
case $g{=}0$ ,

$\mathcal{R}(I)\ =\ M(0)^{2}\ .$ (21)

A novel feature appears for integral values of the coupling, say
$g=2,3,4,\ldots$. Shifting it all the way from $1{-}g$ to $g$, the combined
intertwiner

$Q(g)\ =\ M(g{-}1)M(g{-}2)\cdots M(1)M(0)M(-1)\cdots M(2{-}g)M(1{-}g)$ (22)

also commutes with all Liouville charges but, as a product of an odd number of
intertwiners, it is functionally independent. Only its square belongs to the
ring of Liouville charges,

$Q(g)^{2}\ =\ \mathcal{R}(I(g))^{2g-1}\ .$ (23)

Adjoining $Q(g)$ to the $2N{-}1$ conserved quantities $\\{I_{k},F_{k}\\}$
provides a $\mathbb{Z}_{2}$ grading and makes our model analytically
integrable, with $2N$ independent integrals of motion. We suspect other $Q$
intertwiners based on different shift operators $M$ to be functionally
dependent. The full set of commutators (also with the $F_{\ell}$) is given in
[7] for the $A_{n-1}\oplus A_{1}$ root system.

In the non-simply-laced case, one expects to find polynomials
$\tau^{\prime}_{m^{\prime}}$ antisymmetric under short-root reflections but
symmetric under long-root ones, as well as polynomials $\tau_{m}$ with the
opposite behavior. Inserting Dunkl operators as arguments and performing the
symmetric restriction, we produce $I_{k}$ intertwiners $M_{s}(g_{s},g_{\ell})$
and $M_{\ell}(g_{s},g_{\ell})$, which shift by unity only one coupling but not
the other. Also here, those shift operators allow one to build the joint
$I_{k}$ eigenstates for $(g_{s},g_{\ell})\in\mathbb{Z}\times\mathbb{Z}$ by
repeated application on the free eigenstates. Since we can independently
“wrap” from $1{-}g$ to $g$ for the short roots or for the long roots, there
exist two grading operators, $Q_{s}$ and $Q_{\ell}$, which we expect to be
functionally independent of one another.

##  2 The $\bf{A_{1}\oplus A_{2}}$ model

###  2.1 Integrals of motion

This is the traditional rational three-particle Calogero model, enhanced by an
external inverse-square potential for the center of mass. It is reducible
because the center-of-mass coordinate and momentum,

$X\ =\ {\textstyle\frac{1}{3}}(x^{1}+x^{2}+x^{3}){\qquad{\rm and}\qquad}P\ =\
p_{1}+p_{2}+p_{3}\ ,$ (24)

can be separated from the other degrees of freedom. Therefore, we may
introduce two coupling constants, say $g$ and $g^{\prime}$. The Hamiltonian of
the system (2) is given by ($i,j=1,2,3$)

$\displaystyle H$ $\displaystyle\ =\ \tfrac{1}{2}\sum_{i}p_{i}^{2}\ +\
\sum_{i<j}\frac{g(g{-}1)}{(x^{i}{-}x^{j})^{2}}\ +\
\frac{3\,g^{\prime}(g^{\prime}{-}1)}{2(x^{1}{+}x^{2}{+}x^{3})^{2}}$ (25)
$\displaystyle\ =\ \tfrac{1}{6}P^{2}\ +\
\frac{g^{\prime}(g^{\prime}{-}1)}{6\,X^{2}}\ +\
\tfrac{1}{6}\sum_{i<j}(p_{i}{-}p_{j})^{2}\ +\
\sum_{i<j}\frac{g(g{-}1)}{(x^{i}{-}x^{j})^{2}}\ =\ H_{1}\ +\ H_{2}\ .$

One can choose the positive roots as

$\mathcal{R}_{+}\ =\ \\{e_{1}{-}e_{2},\hskip 5.69046pte_{1}{-}e_{3},\hskip
5.69046pte_{2}{-}e_{3},\hskip 5.69046pte_{1}{+}e_{2}{+}e_{3}\\},$ (26)

such that the Dunkl operators (1) read

$\mathcal{D}_{i}\ =\ \partial_{i}\ -\ \sum_{j(\neq
i)}\frac{g}{x^{i}{-}x^{j}}s_{i-j}\ -\
\frac{g^{\prime}}{x^{1}{+}x^{2}{+}x^{3}}s_{0}\ ,$ (27)

where the $s_{i-j}$ are the two-particle permutation operators,

$\displaystyle s_{1-2}$ $\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\
(x^{2},x^{1},x^{3})\ ,$ (28) $\displaystyle s_{1-3}$ $\displaystyle:\
(x^{1},x^{2},x^{3})\ \mapsto\ (x^{3},x^{2},x^{1})\ ,$ $\displaystyle s_{2-3}$
$\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\ (x^{1},x^{3},x^{2})\ ,$
$\displaystyle\textrm{and}\quad s_{0}$ $\displaystyle:\ (x^{1},x^{2},x^{3})\
\mapsto\ (x^{1},x^{2},x^{3})-2X(1,1,1)\ .$

The lowest three Weyl-invariant polynomials are

$\displaystyle\sigma_{2}(x)$ $\displaystyle\ =\
(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}\ ,$ (29)
$\displaystyle\tilde{\sigma}_{2}(x)$ $\displaystyle\ =\
(x^{1}{-}x^{2})^{2}+(x^{2}{-}x^{3})^{2}+(x^{3}{-}x^{1})^{2}\ ,$
$\displaystyle\tilde{\sigma}_{3}(x)$ $\displaystyle\ =\
(x^{1}{+}x^{2}{-}2x^{3})(x^{2}{+}x^{3}{-}2x^{1})(x^{3}{+}x^{1}{-}2x^{2})\ ,$

where the center-of-mass coordinate is only contained in $\sigma_{2}$.

In this basis, the first three Liouville integrals read

$\displaystyle I_{2}$ $\displaystyle\ =\ -{\rm
res}\bigl{(}\mathcal{D}_{1}^{2}{+}\mathcal{D}_{2}^{2}{+}\mathcal{D}_{3}^{2}\bigr{)}\
=\ 2\,H\ ,$ (30) $\displaystyle\tilde{I}_{2}$ $\displaystyle\ =\ -{\rm
res}\bigl{(}(\mathcal{D}_{1}{-}\mathcal{D}_{2})^{2}+(\mathcal{D}_{2}{-}\mathcal{D}_{3})^{2}+(\mathcal{D}_{3}{-}\mathcal{D}_{1})^{2}\bigr{)}\
=\ 6\,H_{2}\ ,$ $\displaystyle\tilde{I}_{3}$ $\displaystyle\ =\ \
{\mathrm{i}}\,{\rm
res}\bigl{(}(\mathcal{D}_{1}{+}\mathcal{D}_{2}{-}2\mathcal{D}_{3})(\mathcal{D}_{2}{+}\mathcal{D}_{3}{-}2\mathcal{D}_{1})(\mathcal{D}_{3}{+}\mathcal{D}_{1}{-}2\mathcal{D}_{2})\bigr{)}$
$\displaystyle\ =\
\prod_{i=1}^{3}(P{-}3p_{i})-9g(g{-}1)\left(\frac{(P{-}3p_{3})}{(x^{1}{-}x^{2})^{2}}+\frac{(P{-}3p_{2})}{(x^{3}{-}x^{1})^{2}}+\frac{(P{-}3p_{1})}{(x^{2}{-}x^{3})^{2}}\right)\
,$

and they are functionally independent and in involution,

$[I_{2},\tilde{I}_{2}]=[I_{2},\tilde{I}_{3}]=[\tilde{I}_{2},\tilde{I}_{3}]=0\
.$ (31)

The two lowest additional integrals of motion, which are not in involution,
are

$\tilde{F}_{2}\ =\
\\{H,\tilde{J}_{2}\\}-{{\textstyle\frac{1}{2}}}\\{\tilde{I}_{2},D\\}{\qquad{\rm
and}\qquad}\tilde{F}_{3}\ =\ \\{H,\tilde{J}_{3}\\}-\
{{\textstyle\frac{1}{2}}}\\{\tilde{I}_{3},D\\}\ ,\\\ $ (32)

where

$\displaystyle\tilde{J}_{2}$ $\displaystyle\ =\
-\tfrac{3}{2}\sum_{i=1}^{3}\\{X{-}x^{i},p_{i}\\}\ ,$ (33)
$\displaystyle\tilde{J}_{3}$ $\displaystyle\ =\
\sum_{i=1}^{3}(P{-}3p_{i})(X{-}x^{i})(P{-}3p_{i})-9g(g{-}1)\left(\frac{X{-}x^{1}}{(x^{2}{-}x^{3})^{2}}+\frac{X{-}x^{2}}{(x^{3}{-}x^{1})^{2}}+\frac{X{-}x^{3}}{(x^{1}{-}x^{2})^{2}}\right)\
.$

###  2.2 Energy eigenstates

The Hamiltonian eigenfunctions are for this case are

$\Psi^{(g,g^{\prime})}_{{\scriptscriptstyle E},\ell_{2},\ell_{3}}(x)\ \equiv\
{\langle}x\mid\ell_{2},\ell_{3}{\rangle}_{g,g^{\prime}}\ =\
j_{q}({\scriptstyle\sqrt{2E}}\,r)\,r^{-q}{\Delta\vphantom{\big{|}}}^{g}\,{X\vphantom{\big{|}}}^{g^{\prime}}\,h^{(g,g^{\prime})}_{\ell_{2},\ell_{3}}(x)\qquad\textrm{with}\qquad
q=3g+g^{\prime}+2\ell_{2}+3\ell_{3}\ ,$ (34)

where $\Delta=(x^{1}{-}x^{2})(x^{2}{-}x^{3})(x^{3}{-}x^{1})$ is the basic
anti-invariant, $j_{q}$ denotes the spherical Bessel function, and

$\displaystyle h_{\ell_{2},\ell_{3}}^{(g,g^{\prime})}(x)$ $\displaystyle\sim\
r^{6g+2g^{\prime}+1+4\ell_{2}+6\ell_{3}}\,{\Delta\vphantom{\big{|}}}^{-g}\,{X\vphantom{\big{|}}}^{-g^{\prime}}\,\tilde{\sigma}_{2}({\cal
D})^{\ell_{2}}\,\tilde{\sigma}_{3}({\cal
D})^{\ell_{3}}\,{X\vphantom{\big{|}}}^{g^{\prime}}\,{\Delta\vphantom{\big{|}}}^{g}\,r^{-1-6g-2g^{\prime}}$
(35) $\displaystyle\sim\
r^{6g+2g^{\prime}+1+4\ell_{2}+6\ell_{3}}\,\tilde{\sigma}_{2}(\widetilde{{\cal
D}})^{\ell_{2}}\,\tilde{\sigma}_{3}(\widetilde{{\cal
D}})^{\ell_{3}}\,r^{-1-6g-2g^{\prime}}$

is a deformed harmonic polynomial of degree $2\ell_{2}{+}3\ell_{3}$.
Conjugation with
${X\vphantom{\big{|}}}^{g^{\prime}}{\Delta\vphantom{\big{|}}}^{g}$ defines the
“potential-free” Dunkl operators

$\widetilde{{\cal D}}_{i}\ =\ \partial_{i}\ +\sum_{\alpha\in
R^{+}}\dfrac{g_{\alpha}\alpha_{i}}{(\alpha,x)}(1{-}s_{\alpha})\ =\
\partial_{i}\ +\sum_{j(\neq i)}\frac{g}{x^{i}{-}x^{j}}(1{-}s_{i-j})\ +\
\frac{g^{\prime}}{x^{1}{+}x^{2}{+}x^{3}}(1{-}s_{0})\ .$ (36)

The first polynomials read (up to a normalization constant)

$\displaystyle h^{(g,g^{\prime})}_{0,0}(x)$ $\displaystyle\ =\ 1\ ,$ (37)
$\displaystyle h^{(g,g^{\prime})}_{1,0}(x)$ $\displaystyle\ =\
6(3g{+}1)\sigma_{2}-(6g{+}2g^{\prime}{+}3)\tilde{\sigma}_{2}\ ,$
$\displaystyle h^{(g,g^{\prime})}_{0,1}(x)$ $\displaystyle\ =\
\tilde{\sigma}_{3}\ ,$ $\displaystyle h^{(g,g^{\prime})}_{2,0}(x)$
$\displaystyle\ =\
36(3g{+}1)(3g{+}2)\sigma_{2}^{2}+(6g{+}2g^{\prime}{+}5)(6g+2g^{\prime}{+}7)\tilde{\sigma}_{2}^{2}-12(3g{+}2)(6g{+}2g^{\prime}{+}5)\sigma_{2}\tilde{\sigma}_{2}\
,$ $\displaystyle h^{(g,g^{\prime})}_{1,1}(x)$ $\displaystyle\ =\
(6g{+}2g^{\prime}{+}9)\tilde{\sigma}_{2}\tilde{\sigma}_{3}-6(3g{+}4)\sigma_{2}\tilde{\sigma}_{3}\
,$ $\displaystyle h^{(g,g^{\prime})}_{0,2}(x)$ $\displaystyle\ =\
\kappa_{1}\tilde{\sigma}_{2}^{2}\sigma_{2}+\kappa_{2}\tilde{\sigma}_{2}\sigma_{2}^{2}+\kappa_{3}\sigma_{2}^{3}+\kappa_{4}\tilde{\sigma}_{3}^{2}\
,$ $\displaystyle h^{(g,g^{\prime})}_{3,0}(x)$ $\displaystyle\ =\
\kappa_{5}\sigma_{2}^{3}+\kappa_{6}\tilde{\sigma}_{2}\sigma_{2}^{2}+\kappa_{7}\tilde{\sigma}_{2}^{2}\sigma_{2}+\kappa_{8}\tilde{\sigma}_{2}^{3}\
,$ $\displaystyle h^{(g,g^{\prime})}_{2,1}(x)$ $\displaystyle\ =\
\kappa_{9}\tilde{\sigma}_{3}\sigma_{2}^{2}+\kappa_{10}\tilde{\sigma}_{2}\tilde{\sigma}_{3}\sigma_{2}+\kappa_{11}\tilde{\sigma}_{2}^{2}\tilde{\sigma}_{3}\
,$

where the coefficients $\kappa_{i}$ are given in Appendix A.

###  2.3 Intertwining Operators

The $A_{1}{\oplus}A_{2}$ model features two independent anti-invariant
polynomials,

$\tau^{\prime}_{1}(x)\ =\ x^{1}{+}x^{2}{+}x^{3}\ =\ 3\,X{\qquad{\rm
and}\qquad}\tau_{3}(x)\ =\ (x^{1}{-}x^{2})(x^{2}{-}x^{3})(x^{3}{-}x^{1})\ =\
{\Delta\vphantom{\big{|}}}\ ,$ (38)

where the first one is invariant under $s_{i-j}$ and anti-invariant under
$s_{0}$, and the second one behaves oppositely. They lead to the two
intertwining operators

$M^{\prime}(g,g^{\prime})\ =\
\text{res}\bigl{(}\mathcal{D}_{1}{+}\mathcal{D}_{2}{+}\mathcal{D}_{3}\bigr{)}{\qquad{\rm
and}\qquad}M(g,g^{\prime})\ =\
\text{res}\bigl{(}(\mathcal{D}_{1}{-}\mathcal{D}_{2})(\mathcal{D}_{2}{-}\mathcal{D}_{3})(\mathcal{D}_{3}{-}\mathcal{D}_{1})\bigr{)}$
(39)

satisfying (17) for one of the couplings but not shifting the other one. For
the first intertwiner one easily finds
$M^{\prime}(g,g^{\prime})=\sum_{i}\partial_{i}-g^{\prime}/X$ independent of
$g$. The second intertwiner is a more complicated expression; the explicit
form of $M(g,g^{\prime}{=}0)$ is given in [7]. The operators $\mathcal{R}(I)$
(20) associated to $M$ are obtained from the free case ($g{=}g^{\prime}{=}0$),

$\displaystyle\mathcal{R}(I)$ $\displaystyle\ =\ M(0,0)^{2}\ =\
(\partial_{1}{-}\partial_{2})^{2}(\partial_{2}{-}\partial_{3})^{2}(\partial_{3}{-}\partial_{1})^{2}$
(40) $\displaystyle\ =\
-\tfrac{1}{27}\bigl{(}(\partial_{1}{+}\partial_{2}{-}2\partial_{3})(\partial_{2}{+}\partial_{3}{-}2\partial_{1})(\partial_{3}{+}\partial_{1}{-}2\partial_{2})\bigr{)}^{2}+\tfrac{1}{54}\bigl{(}(\partial_{1}{-}\partial_{2})^{2}+(\partial_{2}{-}\partial_{3})^{2}+(\partial_{3}{-}\partial_{1})^{2}\bigr{)}^{3}$
$\displaystyle\ =\
\tfrac{1}{27}\tilde{I}_{3}^{2}-\tfrac{1}{54}\tilde{I}_{2}^{3}\ .$

For the $g^{\prime}$ intertwiner $M^{\prime}$, we have

$\mathcal{R^{\prime}}(I)\ =\ M^{\prime}(0,0)^{2}\ =\
(\partial_{1}{+}\partial_{2}{+}\partial_{3})^{2}\ =\
3(\partial_{1}^{2}{+}\partial_{2}^{2}{+}\partial_{3}^{2})-(\partial_{1}{-}\partial_{2})^{2}-(\partial_{2}{-}\partial_{3})^{2}-(\partial_{3}{-}\partial_{1})^{2}\
=\ \tilde{I}_{2}{-}3\,I_{2}\ .$ (41)

Applying a ($g{-}1$)-fold Darboux dressing with $M(h,0)$ for $h=1,2,\ldots
g{-}1$ and $M(h,0)^{*}=M(-h,0)$ to

$Q(1,0)\ =\ M(0,0)\ =\
(\partial_{1}{-}\partial_{2})(\partial_{2}{-}\partial_{3})(\partial_{3}{-}\partial_{1})\
,$ (42)

we obtain an exceptional independent conserved charge $Q(g,0)$ in the case of
integer coupling $g$ and $g^{\prime}{=}0$. Likewise, Darboux dressing with
$M^{\prime}(0,h^{\prime})$ and
$M^{\prime}(0,h^{\prime})^{*}=M^{\prime}(0,-h^{\prime})$ of

$Q^{\prime}(0,1)\ =\ M^{\prime}(0,0)\ =\
\partial_{1}{+}\partial_{2}{+}\partial_{3}$ (43)

produces such a charge $Q^{\prime}(0,g^{\prime})$ for $g{=}0$ and integer
$g^{\prime}$. Combining both, by following a sequence of $M$ and $M^{\prime}$
intertwiners starting either from $(1{-}g,g^{\prime})$ to $(g,g^{\prime})$ or
from $(g,1{-}g^{\prime})$ to $(g,g^{\prime})$, we can extend these special
charges to $Q(g,g^{\prime})$ and $Q^{\prime}(g,g^{\prime})$, respectively, for
all integral values of both couplings. In such cases, the two extra charges

$\displaystyle Q(g,g^{\prime})$ $\displaystyle\ =\
M(g{-}1,g^{\prime})\,M(g{-}2,g^{\prime})\cdots
M(1,g^{\prime})\,M(0,g^{\prime})\,M(-1,g^{\prime})\cdots
M(2{-}g,g^{\prime})\,M(1{-}g,g^{\prime})\ ,$ (44) $\displaystyle
Q^{\prime}(g,g^{\prime})$ $\displaystyle\ =\
M^{\prime}(g,g^{\prime}{-}1)\,M^{\prime}(g,g^{\prime}{-}2)\cdots
M^{\prime}(g,1)\,M^{\prime}(g,0)\,M^{\prime}(g,-1)\cdots
M^{\prime}(g,2{-}g^{\prime})\,M^{\prime}(g,1{-}g^{\prime})$

enhance the nonlinear algebra of integrals of motion to a
$\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$ graded one,

$\displaystyle{\mathrm{i}}[\tilde{I}_{2},\tilde{F}_{2}]\ =\
2(3\tilde{I}_{2}I_{2}-\tilde{I}_{2}^{2})\
,\quad{\mathrm{i}}[\tilde{I}_{3},\tilde{F}_{2}]\ =\
3(3\tilde{I}_{3}I_{2}-\tilde{I}_{2}\tilde{I}_{3})\ ,$ (45)
$\displaystyle{\mathrm{i}}[\tilde{I}_{2},\tilde{F}_{3}]\ =\
2(3\tilde{I}_{3}I_{2}-\tilde{I}_{3}\tilde{I}_{2})\
,\quad{\mathrm{i}}[\tilde{I}_{3},\tilde{F}_{3}]\ =\
3(-\tilde{I}_{3}^{2}+{\textstyle\frac{3}{2}}\tilde{I}_{2}^{2}I_{2})\ ,$
$\displaystyle{\mathrm{i}}[\tilde{F}_{2},\tilde{F}_{3}]\ =\
\tfrac{1}{2}(\tilde{F}_{3}\tilde{I}_{2}+\tilde{I}_{2}\tilde{F}_{3})-\tfrac{3}{2}(\tilde{F}_{3}I_{2}+I_{2}\tilde{F}_{3})\
,$ $\displaystyle{\mathrm{i}}[Q,\tilde{F}_{2}]\ =\
-3(2g{-}1)Q(\tilde{I}_{2}-3I_{2})\ ,\quad{\mathrm{i}}[Q,\tilde{F}_{3}]\ =\
-3(2g{-}1)Q\tilde{I}_{3}\ ,$
$\displaystyle{\mathrm{i}}[Q^{\prime},\tilde{F}_{2}]\ =\
-(2g^{\prime}{-}1)Q^{\prime}\tilde{I}_{2}\
,\quad{\mathrm{i}}[Q^{\prime},\tilde{F}_{3}]\ =\
-(2g^{\prime}{-}1)Q^{\prime}\tilde{I}_{3}\ ,$ $\displaystyle Q^{2}\ =\
(\mathcal{R}(I))^{2g-1}\ ,\quad Q^{\prime 2}\ =\
(\mathcal{R^{\prime}}(I))^{2g^{\prime}-1}\ ,\quad[Q,Q^{\prime}]\ =\ 0\ .$

###  2.4 Translation-invariant limit

It is instructive to consider the special case of $g^{\prime}{=}0$, where
translation invariance is recovered and the center of mass feels no potential.
In this limit, the reflection $s_{0}$ disappears from the consideration, and
the total momentum appears as a first-order conserved charge. It is then more
convenient to replace (29) with the Newton sums

$\displaystyle\sigma_{1}(x)$ $\displaystyle\ =\ x^{1}+x^{2}+x^{3}\ =\ 3\,X\ ,$
(46) $\displaystyle\sigma_{2}(x)$ $\displaystyle\ =\
(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}\ ,$ $\displaystyle\sigma_{3}(x)$
$\displaystyle\ =\ (x^{1})^{3}+(x^{2})^{3}+(x^{3})^{3}\ ,$

resulting in the basic Liouville integrals

$\displaystyle I_{1}$ $\displaystyle\ =\ -{\mathrm{i}}\,{\rm
res}\bigl{(}\mathcal{D}_{1}{+}\mathcal{D}_{2}{+}\mathcal{D}_{3}\bigr{)}\ =\ P\
,$ (47) $\displaystyle I_{2}$ $\displaystyle\ =\ \ -{\rm
res}\bigl{(}\mathcal{D}_{1}^{2}{+}\mathcal{D}_{2}^{2}{+}\mathcal{D}_{3}^{2}\bigr{)}\
=\ 2\,H\ ,$ $\displaystyle I_{3}$ $\displaystyle\ =\ \ \ {\mathrm{i}}\,{\rm
res}\bigl{(}\mathcal{D}_{1}^{3}{+}\mathcal{D}_{2}^{3}{+}\mathcal{D}_{3}^{3}\bigr{)}\
=\
\sum_{i}p_{i}^{3}+3\sum_{i<j}^{3}\frac{g(g{-}1)}{(x^{i}{-}x^{j})^{2}}(p_{i}{+}p_{j})\
.$

In this case, the two further integrals establishing superintegrability may be
chosen as

$F_{1}\ =\ \\{H,3X\\}-{{\textstyle\frac{1}{2}}}\\{P,D\\}{\qquad{\rm
and}\qquad}F_{3}\ =\ \\{H,J_{3}\\}-{{\textstyle\frac{1}{2}}}\\{I_{3},D\\}\ ,$
(48)

with

$J_{3}\ =\ p_{1}x^{1}p_{1}{+}p_{2}x^{2}p_{2}{+}p_{3}x^{3}p_{3}\ +\
g(g{-}1)\bigl{(}\tfrac{x^{1}+x^{2}}{(x^{1}-x^{2})^{2}}+\tfrac{x^{2}+x^{3}}{(x^{2}-x^{3})^{2}}+\tfrac{x^{3}+x^{1}}{(x^{3}-x^{1})^{2}}\bigr{)}\
.$ (49)

Due to the new first-order charge $P$, the energy eigenfunctions carry a label
$\ell_{1}$ rather than $\ell_{2}$,

$\Psi^{(g)}_{{\scriptscriptstyle E},\ell_{1},\ell_{3}}(x)\ =\
j_{q}({\scriptstyle\sqrt{2E}}\,r)\,r^{-q}{\Delta\vphantom{\big{|}}}^{g}\,h^{(g)}_{\ell_{1},\ell_{3}}(x)\qquad\textrm{with}\qquad
q=3g+\ell_{1}+3\ell_{3}\ ,$ (50)

where now

$\displaystyle h_{\ell_{1},\ell_{3}}^{(g)}(x)$ $\displaystyle\sim\
r^{6g+1+2\ell_{1}+6\ell_{3}}\,{\Delta\vphantom{\big{|}}}^{-g}\,\bigl{(}{\cal
D}_{1}{+}{\cal D}_{2}{+}{\cal D}_{3}\bigr{)}^{\ell_{1}}\,\bigl{(}{\cal
D}_{1}^{3}{+}{\cal D}_{2}^{3}{+}{\cal
D}_{3}^{3}\bigr{)}^{\ell_{3}}\,{\Delta\vphantom{\big{|}}}^{g}\,r^{-1-6g}$ (51)
$\displaystyle\sim\ r^{6g+1+2\ell_{1}+6\ell_{3}}\,\bigl{(}\widetilde{{\cal
D}}_{1}{+}\widetilde{{\cal D}}_{2}{+}\widetilde{{\cal
D}}_{3}\bigr{)}^{\ell_{1}}\,\bigl{(}\widetilde{{\cal
D}}_{1}^{3}{+}\widetilde{{\cal D}}_{2}^{3}{+}\widetilde{{\cal
D}}_{3}^{3}\bigr{)}^{\ell_{3}}\,r^{-1-6g}$

is a deformed harmonic polynomial of degree $\ell_{1}{+}3\ell_{3}$. The first
polynomials can be written as

$\displaystyle h_{0,0}^{(g)}$ $\displaystyle=1\ ,$ (52) $\displaystyle
h_{1,0}^{(g)}$ $\displaystyle=\sigma_{1}\ ,$ $\displaystyle h_{2,0}^{(g)}$
$\displaystyle=(2g{+}1)\sigma_{1}^{2}{-}\sigma_{2}\ ,$ $\displaystyle
h_{0,1}^{(g)}$
$\displaystyle=3(2g{+}1)\sigma_{1}\sigma_{2}{-}(6g{+}5)\sigma_{3}\ ,$
$\displaystyle h_{3,0}^{(g)}$
$\displaystyle=-(6g{+}5)\sigma_{1}^{3}{+}9\sigma_{2}\sigma_{1}\ ,$
$\displaystyle h_{1,1}^{(g)}$
$\displaystyle=-3(2g{+}1)(6g{+}5)\sigma_{2}\sigma_{1}^{2}{+}(6g{+}5)(6g{+}7)\sigma_{3}\sigma_{1}{-}6\sigma_{2}^{2}\
,$ $\displaystyle h_{4,0}^{(g)}$
$\displaystyle=(6g{+}5)(6g{+}7)\sigma_{1}^{4}{-}18(6g{+}5)\sigma_{2}\sigma_{1}^{2}{+}27\sigma_{2}^{2}\
,$ $\displaystyle h_{2,1}^{(g)}$
$\displaystyle=(2g{+}1)(6g{+}7)\sigma_{2}\sigma_{1}^{3}{-}(6g{+}7)(2g{+}3)\sigma_{3}\sigma_{1}^{2}{-}3(2g{-}1)\sigma_{2}^{2}\sigma_{1}{+}(6g{+}7)\sigma_{2}\sigma_{3}\
,$ $\displaystyle h_{5,0}^{(g)}$
$\displaystyle=-(6g{+}7)(2g{+}3)\sigma_{1}^{5}+10(6g{+}7)\sigma_{2}\sigma_{1}^{3}-45\sigma_{2}^{2}\sigma_{1}\
,$ $\displaystyle h_{0,2}^{(g)}$
$\displaystyle=\gamma_{1}\sigma_{2}^{2}\sigma_{1}^{2}{+}\gamma_{2}\sigma_{2}^{3}+\gamma_{3}\sigma_{2}\sigma_{1}^{4}+\gamma_{4}\sigma_{2}\sigma_{3}\sigma_{1}+\gamma_{5}\sigma_{3}^{2}\
,$ $\displaystyle h_{3,1}^{(g)}$
$\displaystyle=\gamma_{6}\sigma_{2}\sigma_{1}^{4}+\gamma_{7}\sigma_{3}\sigma_{1}^{3}+\gamma_{8}\sigma_{2}^{2}\sigma_{1}^{2}+\gamma_{9}\sigma_{2}\sigma_{3}\sigma_{1}+30\sigma_{2}^{3}\
,$ $\displaystyle h_{6,0}^{(g)}$
$\displaystyle=\gamma_{10}\sigma_{1}^{6}+\gamma_{11}\sigma_{2}\sigma_{1}^{4}+\
\gamma_{12}\sigma_{2}^{2}\sigma_{1}^{2}-135\sigma_{2}^{3}\ ,$

where the coefficients $\gamma_{i}$ are given in Appendix A. Roughly half of
these polynomials (as linear combinations) can be obtained as the translation-
invariant limit $h_{\ell_{2},\ell_{3}}^{(g,g^{\prime}=0)}$ of the $A_{1}\oplus
A_{2}$ polynomials, but new states arise in the limit, due to the new first-
order invariant $\sigma_{1}$.

Only the third-order intertwiner $M(g){\equiv}M(g,0)$ based on
${\Delta\vphantom{\big{|}}}$ remains, and via Darboux dressing it generates
the extra charge $Q(g){\equiv}Q(g,0)$ for integral values of $g$, which
enlarges the nonlinear algebra spanned by
$\\{I_{1},I_{2},I_{3},F_{1},F_{3}\\}$ to

$\displaystyle{\mathrm{i}}[I_{1},F_{1}]$ $\displaystyle\ =\ 3I_{2}-I_{1}^{2}\
,\qquad\qquad{\mathrm{i}}[I_{3},F_{1}]\ =\ -3I_{3}I_{1}+3I_{2}^{2}\ ,$ (53)
$\displaystyle{\mathrm{i}}[I_{1},F_{3}]$ $\displaystyle\ =\
-I_{3}I_{1}+I_{2}^{2}\ ,\qquad\quad\\!{\mathrm{i}}[I_{3},F_{3}]\ =\
-3I_{3}^{2}+4I_{3}I_{2}I_{1}+\tfrac{3}{2}I_{2}^{3}-3I_{2}^{2}I_{1}^{2}+\tfrac{1}{2}I_{2}I_{1}^{4}\
,$ $\displaystyle{\mathrm{i}}[F_{1},F_{3}]$ $\displaystyle\ =\
{\textstyle\frac{1}{2}}(F_{1}I_{3}+I_{3}F_{1}+F_{3}I_{1}+I_{1}F_{3})\ ,$
$\displaystyle{\mathrm{i}}[Q,F_{1}]$ $\displaystyle\ =\ -3(2g{-}1)\,Q\,I_{1}\
,\quad\ {\mathrm{i}}[Q,F_{3}]\ =\
-3(2g{-}1)\,Q\bigl{(}I_{3}-\tfrac{2}{3}I_{2}I_{1}\bigr{)}\ ,\qquad Q^{2}\ =\
(\mathcal{R}(I))^{2g-1}\ .$

##  3 The $\bf{AD_{3}}$ model

###  3.1 Integrals of motion

This rank-3 system is irreducible and simply-laced, so contains just a single
coupling $g$. Depending on the choice of variables, it takes the $A_{3}$ or
the $D_{3}$ form. Here, we choose the latter. The Hamiltonian reads

$H\ =\ \tfrac{1}{2}\sum_{i}p_{i}^{2}\
+\sum_{i<j}\Big{(}\frac{g(g{-}1)}{(x^{i}{-}x^{j})^{2}}+\frac{g(g{-}1)}{(x^{i}{+}x^{j})^{2}}\Big{)}\
.$ (54)

One can choose

$\mathcal{R}_{+}=\\{e_{1}{+}e_{2},\hskip 5.69046pte_{1}{+}e_{3},\hskip
5.69046pte_{2}{+}e_{3},\hskip 5.69046pte_{1}{-}e_{2},\hskip
5.69046pte_{1}{-}e_{3},\hskip 5.69046pte_{2}{-}e_{3}\\}\ ,$ (55)

leading to the Dunkl operators

$\mathcal{D}_{i}\ =\ \partial_{i}\ -\sum_{j(\neq
i)}\Big{(}\frac{g}{x^{i}{-}x^{j}}s_{i-j}+\frac{g}{x^{i}{+}x^{j}}s_{i+j}\Big{)}$
(56)

with reflections $s_{i-j}$ given in (28) and

$\displaystyle s_{1+2}$ $\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\
(-x^{2},-x^{1},x^{3})\ ,$ (57) $\displaystyle s_{1+3}$ $\displaystyle:\
(x^{1},x^{2},x^{3})\ \mapsto\ (-x^{3},x^{2},-x^{1})\ ,$ $\displaystyle
s_{2+3}$ $\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\ (x^{1},-x^{3},-x^{2})\
.$

The lowest three Weyl-invariant polynomials are

$\sigma_{2}(x)\ =\ (x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}\ ,\qquad\sigma_{3}(x)\
=\ x^{1}\,x^{2}\,x^{3}\ ,\qquad\sigma_{4}(x)\ =\
(x^{1})^{4}+(x^{2})^{4}+(x^{3})^{4}\ .$ (58)

Hence, the corresponding Liouville integrals read

$\displaystyle I_{2}$ $\displaystyle\ =\ -{\rm
res}\bigl{(}\mathcal{D}_{1}^{2}{+}\mathcal{D}_{2}^{2}{+}\mathcal{D}_{3}^{2}\bigr{)}\
=\ 2\,H\ ,$ (59) $\displaystyle I_{3}$ $\displaystyle\ =\ \ {\mathrm{i}}\,{\rm
res}\bigl{(}\mathcal{D}_{1}\,\mathcal{D}_{2}\,\mathcal{D}_{3}\bigr{)}\ =\
p_{1}\,p_{2}\,p_{3}-4g(g{-}1)\Bigl{(}\tfrac{x^{1}x^{2}\,p_{3}}{((x^{1})^{2}-(x^{2})^{2})^{2}}+\tfrac{x^{2}x^{3}\,p_{1}}{((x^{2})^{2}-(x^{3})^{2})^{2}}+\tfrac{x^{3}x^{1}\,p_{2}}{((x^{3})^{2}-(x^{1})^{2})^{2}}\Bigr{)}\
,$ $\displaystyle I_{4}$ $\displaystyle\ =\ \ \ {\rm
res}\bigl{(}\mathcal{D}_{1}^{4}{+}\mathcal{D}_{2}^{4}{+}\mathcal{D}_{3}^{4}\bigl{)}\
=\ p_{1}^{4}+2g(g{-}1)\textstyle\sum_{\ell\neq
1}\left\\{p_{1}^{2},{\textstyle\frac{1}{(x^{1}-x^{\ell})^{2}}}{+}{\textstyle\frac{1}{(x^{1}+x^{\ell})^{2}}}\right\\}+16g(g{-}1){\textstyle\frac{x^{1}x^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}p_{1}p_{2}$
$\displaystyle-2{\mathrm{i}}g(g{-}1)\textstyle\sum_{\ell\neq
1}\left\\{p_{1},{\textstyle\frac{1}{(x^{1}-x^{\ell})^{3}}}{+}{\textstyle\frac{1}{(x^{1}+x^{\ell})^{3}}}\right\\}+16g^{2}(g{-}1)^{2}\left({\textstyle\frac{(x^{1})^{4}+(x^{2})^{4}}{((x^{1})^{2}-(x^{2})^{2})^{4}}}+{\textstyle\frac{(x^{1})^{2}+(x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}{\textstyle\frac{(x^{1})^{2}+(x^{3})^{2}}{((x^{1})^{2}-(x^{3})^{2})^{2}}}\right)$
$\displaystyle+\text{cyclic}\ ,$

where the term “cyclic” refers to adding the cyclic permutations of the labels
(1,2,3). The two lowest additional conserved charges are

$F_{3}\ =\ \\{H,J_{3}\\}-{{\textstyle\frac{1}{2}}}\\{I_{3},D\\}{\qquad{\rm
and}\qquad}F_{4}\ =\ \\{H,J_{4}\\}-{{\textstyle\frac{1}{2}}}\\{I_{4},D\\}\ ,$
(60)

with

$\displaystyle J_{3}$ $\displaystyle\ =\
\tfrac{1}{3}(x^{1}p_{2}p_{3}{+}x^{2}p_{3}p_{1}{+}x^{3}p_{1}p_{2})-{\textstyle\frac{4}{3}}g(g{-}1)\Bigl{(}\tfrac{1}{(x^{1}-x^{2})^{2}(x^{1}+x^{2})^{2}}{+}\tfrac{1}{(x^{2}-x^{3})^{2}(x^{2}+x^{3})^{2}}{+}\tfrac{1}{(x^{3}-x^{1})^{2}(x^{3}+x^{1})^{2}}\Bigr{)}x^{1}x^{2}x^{3}\
,$ (61) $\displaystyle J_{4}$ $\displaystyle\ =\
{\textstyle\frac{1}{2}}\\{x^{1},p_{1}^{3}\\}+4g(g{-}1)\left({\textstyle\frac{(x^{1})^{2}+2(x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}+{\textstyle\frac{(x^{1})^{2}+2(x^{3})^{2}}{((x^{1})^{2}-(x^{3})^{2})^{2}}}\right)x^{1}p_{1}-2{\mathrm{i}}g(g{-}1){\textstyle\frac{(x^{1})^{2}+(x^{3})^{2}}{((x^{1})^{2}-(x^{3})^{2})^{2}}}+\text{cyclic}\
.$

###  3.2 Energy eigenstates

For the eigenvalue problem

$\displaystyle H\,\Psi^{(g)}_{{\scriptscriptstyle E},\ell_{3},\ell_{4}}(x)\ =\
E\,\Psi^{(g)}_{{\scriptscriptstyle E},\ell_{3},\ell_{4}}(x)$ (62)

one obtains the energy eigenfunctions

$\Psi^{(g)}_{{\scriptscriptstyle E},\ell_{3},\ell_{4}}(x)\ \equiv\
{\langle}x\mid\ell_{3},\ell_{4}{\rangle}_{g}\ =\
j_{q}({\scriptstyle\sqrt{2E}}\,r)\,r^{-q}{\Delta\vphantom{\big{|}}}^{g}\,h^{(g)}_{\ell_{3},\ell_{4}}(x)\qquad\textrm{with}\qquad
q=6g+3\ell_{3}+4\ell_{4}\ ,$ (63)

where
${\Delta\vphantom{\big{|}}}=(x^{1}{-}x^{2})(x^{1}{+}x^{2})(x^{2}{-}x^{3})(x^{2}{+}x^{3})(x^{3}{-}x^{1})(x^{3}{+}x^{1})$
is the basic anti-invariant, and

$\displaystyle h^{(g)}_{\ell_{3},\ell_{4}}(x)$ $\displaystyle\sim\
r^{12g+1+6\ell_{3}+8\ell_{4}}\,\Delta^{-g}\,\bigl{(}{\cal D}_{1}{\cal
D}_{2}{\cal D}_{3}\bigr{)}^{\ell_{3}}\,\bigl{(}{\cal D}_{1}^{4}{+}{\cal
D}_{2}^{4}{+}{\cal
D}_{3}^{4}\bigr{)}^{\ell_{4}}\,{\Delta\vphantom{\big{|}}}^{g}\,r^{-1-12g}$
(64) $\displaystyle\sim\
r^{12g+1+6\ell_{3}+8\ell_{4}}\,\bigl{(}\widetilde{{\cal
D}}_{1}\widetilde{{\cal D}}_{2}\widetilde{{\cal
D}}_{3}\bigr{)}^{\ell_{3}}\,\bigl{(}\widetilde{{\cal
D}}_{1}^{4}{+}\widetilde{{\cal D}}_{2}^{4}{+}\widetilde{{\cal
D}}_{3}^{4}\bigr{)}^{\ell_{4}}\,r^{-1-12g}$

is a deformed harmonic polynomial of degree $3\ell_{3}{+}4\ell_{4}$.
Conjugation with ${\Delta\vphantom{\big{|}}}^{g}$ defines the “potential-free”
Dunkl operators

$\widetilde{{\cal D}}_{i}\ =\ \ \partial_{i}\ +\sum_{j(\neq
i)}\Big{(}\frac{g}{x^{i}{-}x^{j}}(1{-}s_{i-j})+\frac{g}{x^{i}{+}x^{j}}(1{-}s_{i+j})\Big{)}\
.$ (65)

The first polynomials read

$\displaystyle h^{(g)}_{0,0}(x)$ $\displaystyle\ =\ 1\ ,$ (66) $\displaystyle
h^{(g)}_{1,0}(x)$ $\displaystyle\ =\ \sigma_{3}\ ,$ $\displaystyle
h^{(g)}_{0,1}(x)$ $\displaystyle\ =\
(12g{+}5)\sigma_{4}-(8g{+}3)\sigma_{2}^{2}\ ,$ $\displaystyle
h^{(g)}_{2,0}(x)$ $\displaystyle\ =\
\alpha_{1}\sigma_{2}^{3}+\alpha_{2}\sigma_{3}^{2}+\alpha_{3}\sigma_{2}\sigma_{4}\
,$ $\displaystyle h^{(g)}_{1,1}(x)$ $\displaystyle\ =\
\alpha_{4}\sigma_{2}^{2}\sigma_{3}+\alpha_{5}\sigma_{4}\sigma_{3}\ ,$
$\displaystyle h^{(g)}_{0,2}(x)$ $\displaystyle\ =\
\alpha_{6}\sigma_{2}^{4}+\alpha_{7}\sigma_{2}\sigma_{3}^{2}+\alpha_{8}\sigma_{2}^{2}\sigma_{4}+\alpha_{9}\sigma_{4}^{2}\
,$ $\displaystyle h^{(g)}_{3,0}(x)$ $\displaystyle\ =\
\alpha_{10}\sigma_{3}^{3}+\alpha_{11}\sigma_{2}\sigma_{3}\sigma_{4}+\alpha_{12}\sigma_{2}^{3}\sigma_{3}\
,$ $\displaystyle h^{(g)}_{2,1}(x)$ $\displaystyle\ =\
\alpha_{13}\sigma_{2}^{5}+\alpha_{14}\sigma_{2}^{2}\sigma_{3}^{2}+\alpha_{15}\sigma_{2}^{3}\sigma_{4}+\alpha_{16}\sigma_{3}^{2}\sigma_{4}+\alpha_{17}\sigma_{2}\sigma_{4}^{2}\
,$ $\displaystyle h^{(g)}_{1,2}(x)$ $\displaystyle\ =\
\alpha_{18}\sigma_{2}^{4}\sigma_{3}+\alpha_{19}\sigma_{2}\sigma_{3}^{3}+\alpha_{20}\sigma_{2}^{2}\sigma_{3}\sigma_{4}+\alpha_{21}\sigma_{3}\sigma_{4}^{2}\
,$ $\displaystyle h^{(g)}_{0,3}(x)$ $\displaystyle\ =\
\alpha_{22}\sigma_{2}^{6}+\alpha_{23}\sigma_{2}^{3}\sigma_{3}^{2}+\alpha_{24}\sigma_{2}^{4}\sigma_{4}+\alpha_{25}\sigma_{2}\sigma_{3}^{2}\sigma_{4}+\alpha_{26}\sigma_{2}^{2}\sigma_{4}^{2}+\alpha_{27}\sigma_{4}^{3}\
,$ $\displaystyle h^{(g)}_{4,0}(x)$ $\displaystyle\ =\
\alpha_{28}\sigma_{2}^{6}+\alpha_{29}\sigma_{2}^{3}\sigma_{3}^{2}+\alpha_{30}\sigma_{3}^{4}+\alpha_{31}\sigma_{2}^{4}\sigma_{4}+\alpha_{32}\sigma_{2}\sigma_{3}^{2}\sigma_{4}+\alpha_{33}\sigma_{2}^{2}\sigma_{4}^{2}\
,$

where the coefficients $\alpha_{i}$ are given in Appendix A.

###  3.3 Intertwining Operators

The basic anti-invariant of the $AD_{3}$ model reads

$\tau_{6}(x)\ =\
(x^{1}{-}x^{2})(x^{1}{+}x^{2})(x^{2}{-}x^{3})(x^{2}{+}x^{3})(x^{3}{-}x^{1})(x^{3}{+}x^{1})\
=\ {\Delta\vphantom{\big{|}}}\ ,$ (67)

which produces the intertwiner

$M(g)\ =\
\text{res}\bigl{(}(\mathcal{D}_{1}^{2}{-}\mathcal{D}_{2}^{2})(\mathcal{D}_{2}^{2}{-}\mathcal{D}_{3}^{2})(\mathcal{D}_{3}^{2}{-}\mathcal{D}_{1}^{2})\bigr{)}\
,$ (68)

satisfying (17). The polynomial $\mathcal{R}(I)$ can be computed from the free
case ($g{=}0$),

$\displaystyle\mathcal{R}(I)$ $\displaystyle\ =\ M(0)^{2}\ =\
(\partial_{1}^{2}{-}\partial_{2}^{2})^{2}(\partial_{2}^{2}{-}\partial_{3}^{2})^{2}(\partial_{3}^{2}{-}\partial_{1}^{2})^{2}$
(69) $\displaystyle\ =\
\tfrac{1}{2}I_{4}^{3}-\tfrac{5}{4}I_{4}^{2}I_{2}^{2}-9I_{4}I_{3}^{2}I_{2}+I_{4}I_{2}^{4}-27I_{3}^{4}+5I_{3}^{2}I_{2}^{3}-\tfrac{1}{4}I_{2}^{6}\
.$

When the coupling $g$ takes an integral value, there exists a sixth
independent conserved charge

$Q(g)\ =\ M(g{-}1)\,M(g{-}2)\cdots M(1)\,M(0)\,M(-1)\cdots M(2{-}g)\,M(1{-}g)\
,$ (70)

which extends the nonlinear algebra spanned by
$\\{I_{2},I_{3},I_{4},F_{3},F_{4}\\}$ to

$\displaystyle{\mathrm{i}}[I_{3},F_{3}]$ $\displaystyle\ =\
\tfrac{1}{6}(I_{2}^{3}{-}I_{4}I_{2}{-}18I_{3}^{2})\
,\qquad{\mathrm{i}}[I_{4},F_{3}]\ =\
\tfrac{4}{3}(I_{3}I_{2}^{2}{-}3I_{4}I_{3})\ ,$ (71)
$\displaystyle{\mathrm{i}}[I_{3},F_{4}]$ $\displaystyle\ =\
I_{3}I_{2}^{2}{-}3I_{4}I_{3}\ ,\qquad\qquad\ \ {\mathrm{i}}[I_{4},F_{4}]\ =\
2({-}I_{2}^{4}{+}3I_{4}I_{2}^{2}{+}6I_{3}^{2}I_{2}{-}2I_{4}^{2})\ ,$
$\displaystyle{\mathrm{i}}[F_{3},F_{4}]$ $\displaystyle\ =\
\\{F_{4},I_{3}\\}-\tfrac{1}{2}\\{F_{3},I_{4}{-}I_{2}^{2}\\}\ ,$
$\displaystyle{\mathrm{i}}[Q,F_{3}]$ $\displaystyle\ =\ -6(2g{-}1)\,Q\,I_{3}\
,\quad{\mathrm{i}}[Q,F_{4}]\ =\ 6(2g{-}1)\,Q\,(\tfrac{2}{3}I_{2}^{2}-I_{4})\
,\quad Q^{2}\ =\ (\mathcal{R}(I))^{2g-1}\ .$

##  4 The $\bf{BC_{3}}$ model

###  4.1 Integrals of motion

This is the only irreducible non-simply-laced rank-3 model, so we have to deal
with two coupling constants, $g_{\ell}$ and $g_{s}$. It is described by the
Hamiltonian

$H\ =\ \tfrac{1}{2}\sum_{i}^{3}p_{i}^{2}\
+\sum_{i<j}\Big{(}\frac{g_{\ell}(g_{\ell}{-}1)}{(x^{i}{-}x^{j})^{2}}+\frac{g_{\ell}(g_{\ell}{-}1)}{(x^{i}{+}x^{j})^{2}}\Big{)}\
+\sum_{i}\frac{g_{s}(g_{s}{-}1)}{2\,(x^{i})^{2}}\ .$ (72)

We take the set of positive roots as

$\mathcal{R}_{+}\ =\ \\{e_{1},\hskip 5.69046pte_{2},\hskip
5.69046pte_{3},\hskip 5.69046pte_{1}{+}e_{2},\hskip
5.69046pte_{1}{+}e_{3},\hskip 5.69046pte_{2}{+}e_{3},\hskip
5.69046pte_{1}{-}e_{2},\hskip 5.69046pte_{1}{-}e_{3},\hskip
5.69046pte_{2}{-}e_{3}\\}\ ,$ (73)

so the Dunkl operators read

$\mathcal{D}_{i}\ =\ \partial_{i}\ -\sum_{j(\neq
i)}\Big{(}\frac{g_{\ell}}{x^{i}{-}x^{j}}s_{i-j}+\frac{g_{\ell}}{x^{i}{+}x^{j}}s_{i+j}\Big{)}\
-\ \frac{g_{s}}{x^{i}}s_{i}\ ,$ (74)

where the reflections are given in (28) and (57) and by

$\displaystyle s_{1}$ $\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\
(-x^{1},x^{2},x^{3})\ ,$ (75) $\displaystyle s_{2}$ $\displaystyle:\
(x^{1},x^{2},x^{3})\ \mapsto\ (x^{1},-x^{2},x^{3})\ ,$ $\displaystyle s_{3}$
$\displaystyle:\ (x^{1},x^{2},x^{3})\ \mapsto\ (x^{1},x^{2},-x^{3})\ .$

The three lowest Weyl-invariant polynomials may be chosen as

$\sigma_{2}(x)=(x^{1})^{2}+(x^{2})^{2}+(x^{3})^{2}\
,\qquad\sigma_{4}(x)=(x^{1})^{4}+(x^{2})^{4}+(x^{3})^{4}\
,\qquad\sigma_{6}(x)=(x^{1}\,x^{2}\,x^{3})^{2}=\sigma_{3}^{2}\ ,$ (76)

but one may also remain with the even Newton sums, replacing $\sigma_{6}$ with
$\sigma^{\prime}_{6}(x)=(x^{1})^{6}+(x^{2})^{6}+(x^{3})^{6}$.

For the basis (76), the first three Liouville integrals take the form

$\displaystyle I_{2}$ $\displaystyle\ =\ -{\rm
res}\bigl{(}\mathcal{D}_{1}^{2}{+}\mathcal{D}_{2}^{2}{+}\mathcal{D}_{3}^{2}\bigr{)}\
=\ 2\,H\ ,$ (77) $\displaystyle I_{4}$ $\displaystyle\ =\ \ \ {\rm
res}\bigl{(}\mathcal{D}_{1}^{4}{+}\mathcal{D}_{2}^{4}{+}\mathcal{D}_{3}^{4}\bigl{)}\
,$ $\displaystyle I_{6}$ $\displaystyle\ =\ -{\rm
res}\bigl{(}\mathcal{D}_{1}^{2}\,\mathcal{D}_{2}^{2}\,\mathcal{D}_{3}^{2}\bigr{)}\
,$

with the explicit form of $I_{4}$ and $I_{6}$ displayed in Appendix B. Two
additional integrals of motion (not in involution) are

$F_{4}\ =\ \\{H,J_{4}\\}-{{\textstyle\frac{1}{2}}}\\{I_{4},D\\}{\qquad{\rm
and}\qquad}F_{6}\ =\ \\{H,J_{6}\\}-{{\textstyle\frac{1}{2}}}\\{I_{6},D\\}\ ,$
(78)

where $J_{4}$ and $J_{6}$ are also given in Appendix B.

###  4.2 Energy eigenstates

The eigenvalue problem

$\displaystyle H\,\Psi^{(g_{\ell},g_{s})}_{{\scriptscriptstyle
E},\ell_{4},\ell_{6}}(x)\ =\ E\,\Psi^{(g_{\ell},g_{s})}_{{\scriptscriptstyle
E},\ell_{4},\ell_{6}}(x)$ (79)

is solved by

$\Psi^{(g_{\ell},g_{s})}_{{\scriptscriptstyle E},\ell_{4},\ell_{6}}(x)\
\equiv\ {\langle}x\mid\ell_{4},\ell_{6}{\rangle}_{g,g^{\prime}}\ =\
j_{q}({\scriptstyle\sqrt{2E}}\,r)\,r^{-q}\Delta_{\ell}^{g_{\ell}}\Delta_{s}^{g_{s}}\,h^{(g_{\ell},g_{s})}_{\ell_{4},\ell_{6}}(x)\qquad\textrm{with}\qquad
q=6g_{\ell}+3g_{s}+4\ell_{4}+6\ell_{6}\ ,$ (80)

where
$\Delta_{\ell}=(x^{1}{-}x^{2})(x^{1}{+}x^{2})(x^{2}{-}x^{3})(x^{2}{+}x^{3})(x^{3}{-}x^{1})(x^{3}{+}x^{1})$
and $\Delta_{s}=x^{1}\,x^{2}\,x^{3}$, as well as

$\displaystyle h^{(g_{\ell},g_{s})}_{\ell_{4},\ell_{6}}(x)$
$\displaystyle\sim\
r^{12g_{\ell}+6g_{s}+1+8\ell_{4}+12\ell_{6}}\,\Delta_{\ell}^{-g_{\ell}}\Delta_{s}^{-g_{s}}\,\bigl{(}{\cal
D}_{1}^{4}{+}{\cal D}_{2}^{4}{+}{\cal
D}_{3}^{4}\bigr{)}^{\ell_{4}}\,\bigl{(}{\cal D}_{1}{\cal D}_{2}{\cal
D}_{3}\bigr{)}^{2\ell_{6}}\,{\Delta\vphantom{\big{|}}}^{g}\,r^{-1-12g_{\ell}-6g_{s}}$
(81) $\displaystyle\sim\
r^{12g_{\ell}+6g_{s}+1+8\ell_{4}+12\ell_{6}}\,\bigl{(}\widetilde{{\cal
D}}_{1}^{4}{+}\widetilde{{\cal D}}_{2}^{4}{+}\widetilde{{\cal
D}}_{3}^{4}\bigr{)}^{\ell_{4}}\,\bigl{(}\widetilde{{\cal
D}}_{1}\widetilde{{\cal D}}_{2}\widetilde{{\cal
D}}_{3}\bigr{)}^{2\ell_{6}}\,r^{-1-12g_{\ell}-6g_{s}}$

being a deformed harmonic polynomial of degree $4\ell_{4}{+}6\ell_{6}$. The
“potential-free” Dunkl operators read

$\widetilde{{\cal D}}_{i}\ =\ \ \partial_{i}\ +\sum_{j(\neq
i)}\Big{(}\frac{g_{\ell}}{x^{i}{-}x^{j}}(1{-}s_{i-j})+\frac{g_{\ell}}{x^{i}{+}x^{j}}(1{-}s_{i+j})\Big{)}\
+\ \frac{g_{s}}{x^{i}}(1{-}s_{i})\ .$ (82)

The first polynomials read

$\displaystyle h^{(g_{\ell},g_{s})}_{0,0}(x)$ $\displaystyle\ =\ 1\ ,$ (83)
$\displaystyle h^{(g_{\ell},g_{s})}_{1,0}(x)$ $\displaystyle\ =\
-(8g_{\ell}{+}2g_{s}{+}3)\sigma_{2}^{2}+(12g_{\ell}{+}6g_{s}{+}5)\sigma_{4}\
,$ $\displaystyle h^{(g_{\ell},g_{s})}_{0,1}(x)$ $\displaystyle\ =\
\mu_{1}\sigma_{2}^{3}+\mu_{2}\sigma_{2}\sigma_{4}+\mu_{3}\sigma_{6}\ ,$
$\displaystyle h^{(g_{\ell},g_{s})}_{2,0}(x)$ $\displaystyle\ =\
\mu_{4}\sigma_{2}^{4}+\mu_{5}\sigma_{2}^{2}\sigma_{4}+\mu_{6}\sigma_{4}^{2}+\mu_{7}\sigma_{2}\sigma_{6}\
,$ $\displaystyle h^{(g_{\ell},g_{s})}_{1,1}(x)$ $\displaystyle\ =\
\mu_{8}\sigma_{2}^{5}+\mu_{9}\sigma_{2}^{3}\sigma_{4}+\mu_{10}\sigma_{2}\sigma_{4}^{2}+\mu_{11}\sigma_{2}^{2}\sigma_{6}+\mu_{12}\sigma_{4}\sigma_{6}\
,$ $\displaystyle h^{(g_{\ell},g_{s})}_{0,2}(x)$ $\displaystyle\ =\
\mu_{13}\sigma_{2}^{6}+\mu_{14}\sigma_{2}^{4}\sigma_{4}+\mu_{15}\sigma_{2}^{2}\sigma_{4}^{2}+\mu_{16}\sigma_{2}^{3}\sigma_{6}+\mu_{17}\sigma_{2}\sigma_{4}\sigma_{6}+\mu_{18}\sigma_{6}^{2}\
,$ $\displaystyle h^{(g_{\ell},g_{s})}_{3,0}(x)$ $\displaystyle\ =\
\mu_{19}\sigma_{2}^{6}+\mu_{20}\sigma_{2}^{4}\sigma_{4}+\mu_{21}\sigma_{2}^{2}\sigma_{4}^{2}+\mu_{22}\sigma_{2}^{3}\sigma_{6}+\mu_{23}\sigma_{2}\sigma_{4}\sigma_{6}+\mu_{24}\sigma_{4}^{3}\
.$

where the coefficients $\mu_{i}$ are given in Appendix A. It is obvious that
putting $g_{s}=0$ and $g_{\ell}=g$ brings us to the corresponding states (66).

###  4.3 Intertwining Operators

The short-root and long-root anti-invariant polynomials are

$\tau^{\prime}_{3}(x)=x^{1}\,x^{2}\,x^{3}={\Delta\vphantom{\big{|}}}_{s}{\qquad{\rm
and}\qquad}\tau_{6}(x)=(x^{1}{-}x^{2})(x^{1}{+}x^{2})(x^{2}{-}x^{3})(x^{2}{+}x^{3})(x^{3}{-}x^{1})(x^{3}{+}x^{1})={\Delta\vphantom{\big{|}}}_{\ell}\
,$ (84)

yielding the short-root and long-root intertwiners

$M_{s}(g_{\ell},g_{s})\ =\ \text{res}\prod_{i}\mathcal{D}_{i}{\qquad{\rm
and}\qquad}M_{\ell}(g_{\ell},g_{s})\ =\
\text{res}\prod_{i<j}(\mathcal{D}^{2}_{i}{-}\mathcal{D}^{2}_{j})\ ,$ (85)

respectively. They satisfy the relations (17), such that

$\displaystyle M_{s}(g_{\ell},g_{s})\,I_{k}(g_{\ell},g_{s})\ =\
I_{k}(g_{\ell},g_{s}{+}1)\,M_{s}(g_{\ell},g_{s})\ ,$ (86) $\displaystyle
M_{\ell}(g_{\ell},g_{s})\,I_{k}(g_{\ell},g_{s})\ =\
I_{k}(g_{\ell}{+}1,g_{s})\,M_{\ell}(g_{\ell},g_{s})\ .$

From the free case ($g_{\ell}{=}g_{s}{=}0$), one can compute the operators
$\mathcal{R}_{s}(I)$ and $\mathcal{R}_{\ell}(I)$ (20),

$\displaystyle\mathcal{R}_{s}(I)$ $\displaystyle\ =\
\prod_{i}\partial_{i}^{2}\qquad\quad\\!\ =\ -I_{6}\ ,$ (87)
$\displaystyle\mathcal{R}_{\ell}(I)$ $\displaystyle\ =\
\prod_{i<j}(\partial^{2}_{i}{-}\partial^{2}_{j})^{2}\ =\
-27I_{6}^{2}-9I_{6}I_{4}I_{2}+5I_{6}I_{2}^{3}+\tfrac{1}{2}I_{4}^{3}-\tfrac{5}{4}I_{4}^{2}I_{2}^{2}+I_{4}I_{2}^{4}-\tfrac{1}{4}I_{2}^{6}\
.$

Finally, for integral values of both couplings we can construct two more
conserved charges,

$\displaystyle Q_{\ell}(g_{\ell},g_{s})$ $\displaystyle\ =\
M_{\ell}(g_{\ell}{-}1,g_{s})\,\cdots\,M_{\ell}(1,g_{s})\,M_{\ell}(0,g_{s})\,M_{\ell}(-1,g_{s})\,\cdots\,M_{\ell}(1{-}g_{\ell},g_{s})\
,$ (88) $\displaystyle Q_{s}(g_{s},g_{\ell})$ $\displaystyle\ =\
M_{s}(g_{\ell},g_{s}{-}1)\,\cdots\,M_{s}(g_{\ell},1)\,M_{s}(g_{\ell},0)\,M_{s}(g_{\ell},-1)\,\cdots\,M_{s}(g_{\ell},1{-}g_{s})\
.$

Together with $F_{k}$ and $I_{k}$, they satisfy the following nonzero
relations,

$\displaystyle{\mathrm{i}}[I_{4},F_{4}]$ $\displaystyle\ =\
2({-}I_{2}^{4}{+}3I_{4}I_{2}^{2}{+}6I_{6}I_{2}{-}2I_{4}^{2}),\
,\qquad{\mathrm{i}}[I_{6},F_{4}]\ =\ 2(I_{6}I_{2}^{2}{-}3I_{6}I_{4})\ ,$ (89)
$\displaystyle{\mathrm{i}}[I_{4},F_{6}]$ $\displaystyle\ =\
\tfrac{4}{3}I_{6}I_{2}^{2}{-}4I_{6}I_{4}\ ,\qquad{\mathrm{i}}[I_{6},F_{6}]\ =\
\tfrac{1}{3}(I_{6}I_{2}^{3}{-}I_{6}I_{4}I_{2}{-}18I_{6}^{2})\ ,$
$\displaystyle{\mathrm{i}}[F_{4},F_{6}]$ $\displaystyle\ =\
\\{F_{6},2I_{4}{-}I_{2}^{2}\\}-\\{F_{4},I_{6}\\}\ ,$
$\displaystyle{\mathrm{i}}[Q_{s},F_{4}]$ $\displaystyle\ =\
-3(2g_{s}{-}1)Q_{s}(I_{4}{-}{\textstyle\frac{1}{3}}I_{2}^{2})\
,\qquad{\mathrm{i}}[Q_{s},F_{6}]\ =\
-3(2g_{s}{-}1)Q_{s}(I_{6}{+}\tfrac{1}{18}I_{2}I_{4}{-}\tfrac{1}{18}I_{2}^{3})\
,$ $\displaystyle{\mathrm{i}}[Q_{\ell},F_{4}]$ $\displaystyle\ =\
-6(2g_{\ell}{-}1)Q_{\ell}(I_{4}{-}{\textstyle\frac{2}{3}}I_{2}^{2})\
,\qquad{\mathrm{i}}[Q_{\ell},F_{6}]\ =\ -6(2g_{\ell}{-}1)Q_{\ell}I_{6}\ ,$
$\displaystyle Q_{s}^{2}$ $\displaystyle\ =\ (\mathcal{R}_{s}(I))^{2g_{s}-1}\
,\qquad Q_{\ell}^{2}=(\mathcal{R}_{\ell}(I))^{2g_{\ell}-1}\
,\qquad[Q_{s},Q_{\ell}]\ =\ 0\ ,$

which define a $\mathbb{Z}_{2}{\oplus}\mathbb{Z}_{2}$ graded polynomial
algebra of conserved charges.

Acknowledgments: FC-M was partially supported by Fondecyt grant 1171475. FC
was partially supported by Fondecyt grant 1171475 and Becas Santander
Iberoamérica. FC would like to thank the Departamento de Física Teórica,
Átomica y Óptica at the Universidad de Valladolid for all the support and kind
hospitality. OL ist grateful to the Instituto de Ciencias Físicas y
Matemáticas at the Universidad Austral de Chile for warm hospitality.

##  Appendix A Polynomials coefficients

####  $A_{1}\oplus A_{2}$ model

$\displaystyle\kappa_{1}$
$\displaystyle=-27(6g{+}2g^{\prime}{+}7)(6g{+}2g^{\prime}{+}9)\ ,$
$\displaystyle\kappa_{2}$ $\displaystyle=162(3g{+}2)(6g{+}2g^{\prime}{+}7)\ ,$
$\displaystyle\kappa_{3}$ $\displaystyle=-324(3g{+}1)(3g{+}2)\ ,$
$\displaystyle\kappa_{4}$
$\displaystyle=2(6g{+}2g^{\prime}{+}7)(6g{+}2g^{\prime}{+}9)(6g{+}2g^{\prime}{+}11)\
,$ $\displaystyle\kappa_{5}$ $\displaystyle=-648(g{+}1)(3g{+}1)(3g{+}2)\ ,$
$\displaystyle\kappa_{6}$
$\displaystyle=324(g{+}1)(3g{+}2)(6g{+}2g^{\prime}{+}7)\ ,$ (90)
$\displaystyle\kappa_{7}$
$\displaystyle=-54(g{+}1)(6g{+}2g^{\prime}{+}7)(6g{+}2g^{\prime}{+}9)\ ,$
$\displaystyle\kappa_{8}$
$\displaystyle=(6g{+}2g^{\prime}{+}7)(6g{+}2g^{\prime}{+}9)(6g{+}2g^{\prime}{+}11)\
,$ $\displaystyle\kappa_{9}$ $\displaystyle=-36(3g{+}4)(3g{+}5)\ ,$
$\displaystyle\kappa_{10}$ $\displaystyle=12(3g{+}5)(6g{+}2g^{\prime}{+}11)\
,$ $\displaystyle\kappa_{11}$
$\displaystyle=-(6g{+}2g^{\prime}{+}11)(6g{+}2g^{\prime}{+}13)\ .$

####  $A_{2}$ model

$\displaystyle\gamma_{1}$
$\displaystyle=6(6g{+}7)\left(4g^{2}{+}11g{+}10\right)\ ,$
$\displaystyle\gamma_{2}$ $\displaystyle=-3\left(4g^{2}{+}14g{+}17\right)\ ,$
$\displaystyle\gamma_{3}$ $\displaystyle={-}3(2g{+}3)(6g{+}7)\ ,$
$\displaystyle\gamma_{4}$ $\displaystyle={-}12(2g{+}3)^{2}(6g{+}7)\ ,$
$\displaystyle\gamma_{5}$ $\displaystyle=2(2g{+}3)(6g{+}7)(6g{+}11)\ ,$
$\displaystyle\gamma_{6}$ $\displaystyle=-3(2g{+}1)(2g{+}3)(6g{+}7)\ ,$ (91)
$\displaystyle\gamma_{7}$ $\displaystyle=(2g{+}3)(6g{+}7)(6g{+}11)\ ,$
$\displaystyle\gamma_{8}$ $\displaystyle=3(6g{-}1)(6g{+}7)\ ,$
$\displaystyle\gamma_{9}$ $\displaystyle=-9(2g{+}3)(6g{+}7)\ ,$
$\displaystyle\gamma_{10}$ $\displaystyle=(2g{+}3)(6g{+}7)(6g{+}11)\ ,$
$\displaystyle\gamma_{11}$ $\displaystyle=-45(2g{+}3)(6g{+}7)\ ,$
$\displaystyle\gamma_{12}$ $\displaystyle=135(6g{+}7)\ .$

####  $AD_{3}$ model

$\displaystyle\alpha_{1}$ $\displaystyle={-}(28g{+}17)\ ,$
$\displaystyle\alpha_{2}$ $\displaystyle=6(12g{+}7)(12g{+}11)\ ,$
$\displaystyle\alpha_{3}$ $\displaystyle=3(12g{+}7)\ ,$
$\displaystyle\alpha_{4}$ $\displaystyle=(8g{+}5)\ ,$
$\displaystyle\alpha_{5}$ $\displaystyle=-(12g{+}11)\ ,$
$\displaystyle\alpha_{6}$ $\displaystyle=(64g^{2}{+}152g{+}99)\ ,$
$\displaystyle\alpha_{7}$ $\displaystyle=-48(12g{+}13)\ ,$
$\displaystyle\alpha_{8}$ $\displaystyle=-6(32g^{2}{+}76g{+}47)\ ,$
$\displaystyle\alpha_{9}$ $\displaystyle=3(4g{+}5)(12g{+}13)\ ,$
$\displaystyle\alpha_{10}$ $\displaystyle=-2(12g{+}13)(12g{+}17)\ ,$
$\displaystyle\alpha_{11}$ $\displaystyle=-3(12g{+}13)\ ,$
$\displaystyle\alpha_{12}$ $\displaystyle=(28g{+}27)\ ,$
$\displaystyle\alpha_{13}$ $\displaystyle=(224g^{2}{+}492g{+}255)\ ,$
$\displaystyle\alpha_{14}$ $\displaystyle=-6(4g{+}5)(12g{+}11)(24g{+}29)\ ,$
$\displaystyle\alpha_{15}$ $\displaystyle=-4(12g{+}11)(13g{+}18)\ ,$
$\displaystyle\alpha_{16}$ $\displaystyle=6(12g{+}11)(12g{+}17)(12g{+}19)\ ,$
$\displaystyle\alpha_{17}$ $\displaystyle=3(12g{+}11)(12g{+}17)\ ,$
$\displaystyle\alpha_{18}$ $\displaystyle=-(64g^{2}{+}184g{+}159)\ ,$ (92)
$\displaystyle\alpha_{19}$ $\displaystyle=48(12g{+}19)\ ,$
$\displaystyle\alpha_{20}$ $\displaystyle=6(32g^{2}{+}100g{+}83)\ ,$
$\displaystyle\alpha_{21}$ $\displaystyle=-3(4g{+}7)(12g{+}19)\ ,$
$\displaystyle\alpha_{22}$
$\displaystyle=-(73728g^{5}{+}589824g^{4}{+}1898752g^{3}{+}3045328g^{2}{+}2416240g$
$\displaystyle{+}754803)\ ,$ $\displaystyle\alpha_{23}$
$\displaystyle=48(12g{+}13)(12g{+}17)(288g^{2}{+}868g{+}651)\ ,$
$\displaystyle\alpha_{24}$
$\displaystyle=3(12g{+}13)(9216g^{4}{+}63360g^{3}{+}164912g^{2}{+}191336g{+}83097)\
,$ $\displaystyle\alpha_{25}$
$\displaystyle=-432(4g{+}7)(12g{+}13)(12g{+}17)(12g{+}19)\ ,$
$\displaystyle\alpha_{26}$
$\displaystyle=-3(12g{+}13)(12g{+}17)(12g{+}19)(96g^{2}{+}364g{+}357)\ ,$
$\displaystyle\alpha_{27}$
$\displaystyle=3(4g{+}7)(12g{+}13)(12g{+}17)(12g{+}19)(12g{+}23)\ ,$
$\displaystyle\alpha_{28}$
$\displaystyle=(3136g^{3}{+}12656g^{2}{+}16764g{+}7269)\ ,$
$\displaystyle\alpha_{29}$
$\displaystyle=-12(12g{+}13)(12g{+}17)(112g^{2}{+}344g{+}263)\ ,$
$\displaystyle\alpha_{30}$
$\displaystyle=12(4g{+}7)(12g{+}13)(12g{+}17)(12g{+}19)(12g{+}23)\ ,$
$\displaystyle\alpha_{31}$ $\displaystyle=-6(12g{+}13)(112g^{2}{+}336g{+}251)\
,$ $\displaystyle\alpha_{32}$
$\displaystyle=36(4g{+}7)(12g{+}13)(12g{+}17)(12g{+}19)\ ,$
$\displaystyle\alpha_{33}$ $\displaystyle=3(12g{+}13)(12g{+}17)(12g{+}19)\ .$

####  $BC_{3}$ model

$\displaystyle\mu_{1}$ $\displaystyle=-(2g_{s}{+}1)(28g_{\ell}+10g_{s}{+}17)\
,$ $\displaystyle\mu_{2}$ $\displaystyle=3(2g_{s}{+}1)(12g_{\ell}+6g_{s}{+}7)\
,$ $\displaystyle\mu_{3}$
$\displaystyle=6(12g_{\ell}{+}6g_{s}{+}7)(12g_{\ell}{+}6g_{s}{+}11)\ ,$
$\displaystyle\mu_{4}$
$\displaystyle=(64g_{\ell}^{2}{+}4g_{s}^{2}{+}56g_{s}{+}8g_{\ell}(4g_{s}{+}19){+}99)\
,$ $\displaystyle\mu_{5}$
$\displaystyle=-6(32g_{\ell}^{2}{+}4g_{s}^{2}{+}32g_{s}{+}4g_{\ell}(6g_{s}{+}19){+}47)\
,$ $\displaystyle\mu_{6}$
$\displaystyle=3(4g_{\ell}{+}2g_{s}{+}5)(12g_{\ell}{+}6g_{s}{+}13)\ ,$
$\displaystyle\mu_{7}$ $\displaystyle=-48(12g_{\ell}{+}6g_{s}{+}13)\ ,$
$\displaystyle\mu_{8}$
$\displaystyle=(2g_{s}{+}1)(224g_{\ell}^{2}{+}20g_{s}^{2}{+}168g_{s}{+}4g_{\ell}(34g_{s}{+}123){+}255)\
,$ $\displaystyle\mu_{9}$
$\displaystyle=-4(2g_{s}{+}1)(13g_{\ell}{+}4g_{s}{+}18)(12g_{\ell}{+}6g_{s}{+}11)\
,$ $\displaystyle\mu_{10}$
$\displaystyle=3(2g_{s}{+}1)(12g_{\ell}{+}6g_{s}{+}11)(12g_{\ell}{+}6g_{s}{+}17)\
,$ $\displaystyle\mu_{11}$
$\displaystyle=-6(12g_{\ell}{+}6g_{s}{+}11)(96g_{\ell}^{2}{+}12g_{s}^{2}{+}104g_{s}{+}4g_{\ell}(18g_{s}{+}59){+}145)\
,$ $\displaystyle\mu_{12}$
$\displaystyle=6(12g_{\ell}{+}6g_{s}{+}11)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)\
,$ $\displaystyle\mu_{13}$
$\displaystyle=(2g_{s}{+}1)(2g_{s}{+}3)(4g_{\ell}(4191{+}3164g_{\ell}{+}784g_{\ell}^{2}){+}16g_{\ell}g_{s}(645{+}238g_{\ell}{+}95g_{s}){+}2g_{s}(3447{+}1046g_{s}{+}100g_{s}^{2}){+}7269)\
,$ $\displaystyle\mu_{14}$
$\displaystyle=-6(2g_{s}{+}1)(2g_{s}{+}3)(12g_{\ell}{+}6g_{s}{+}13)(112g_{\ell}(g_{\ell}{+}3){+}144g_{s}{+}96g_{\ell}g_{s}{+}20g_{s}^{2}{+}251)\
,$ $\displaystyle\mu_{15}$
$\displaystyle=3(2g_{s}{+}1)(2g_{s}{+}3)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)\
,$ $\displaystyle\mu_{16}$
$\displaystyle=-12(2g_{s}{+}3)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(112g_{\ell}^{2}{+}4g_{s}(5g_{s}{+}38){+}8g_{\ell}(12g_{s}{+}43){+}263)\
,$ $\displaystyle\mu_{17}$
$\displaystyle=36(2g_{s}{+}3)(4g_{\ell}{+}2g_{s}{+}7)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)\
,$ $\displaystyle\mu_{18}$
$\displaystyle=36(4g_{\ell}{+}2g_{s}{+}7)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)(12g_{\ell}{+}6g_{s}{+}23)\
,$ $\displaystyle\mu_{19}$
$\displaystyle=-(73728g_{\ell}^{5}{+}288g_{s}^{5}{+}11952g_{s}^{4}{+}127152g_{s}^{3}{+}552584g_{s}^{2}{+}1064386g_{s}{+}18432g_{\ell}^{4}(7g_{s}{+}32){+}256g_{\ell}^{3}(342g_{s}^{2}{+}3555g_{s}{+}7417)$
$\displaystyle{+}16g_{\ell}^{2}(1800g_{s}^{3}{+}32508g_{s}^{2}{+}146534g_{s}{+}190333){+}16g_{\ell}(288g_{s}^{4}{+}8136g_{s}^{3}{+}59596g_{s}^{2}{+}163350g_{s}{+}151015){+}754803)\
,$

$\displaystyle\mu_{20}$
$\displaystyle=3(12g_{\ell}{+}6g_{s}{+}13)(9216g_{\ell}^{4}{+}144g_{s}^{4}{+}4032g_{s}^{3}{+}30072g_{s}^{2}{+}85520g_{s}{+}1152g_{\ell}^{3}(12g_{s}{+}55){+}16g_{\ell}^{2}(468g_{s}^{2}{+}4824g_{s}{+}10307)$
(93)
$\displaystyle{+}8g_{\ell}(216g_{s}^{3}{+}3852g_{s}^{2}{+}17722g_{s}{+}23917){+}83097)\
,$ $\displaystyle\mu_{21}$
$\displaystyle=-3(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)(96g_{\ell}^{2}{+}12g_{s}^{2}{+}160g_{s}{+}4g_{\ell}(18g_{s}{+}91){+}357)\
,$ $\displaystyle\mu_{22}$
$\displaystyle=48(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(288g_{\ell}^{2}{+}36g_{s}^{2}{+}328g_{s}{+}4g_{\ell}(217{+}54g_{s}){+}651)\
,$ $\displaystyle\mu_{23}$
$\displaystyle=-432(4g_{\ell}{+}2g_{s}{+}7)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)\
,$ $\displaystyle\mu_{24}$
$\displaystyle=3(4g_{\ell}{+}2g_{s}{+}7)(12g_{\ell}{+}6g_{s}{+}13)(12g_{\ell}{+}6g_{s}{+}17)(12g_{\ell}{+}6g_{s}{+}19)(12g_{\ell}{+}6g_{s}{+}23)\
.$

##  Appendix B Formulae for $\bf{BC_{3}}$

Explicitly, the integrals of motion read

$\displaystyle{I_{4}}$ $\displaystyle{\ =\
p_{1}^{4}+g_{s}(g_{s}{-}1)\\{p_{1}^{2},{\textstyle\frac{1}{(x^{1})^{2}}}\\}+g_{s}^{2}(g_{s}{-}1)^{2}{\textstyle\frac{1}{(x^{1})^{4}}}+2g_{\ell}(g_{\ell}{-}1)\textstyle\sum_{\ell\neq
1}^{3}\bigl{\\{}p_{1}^{2},{\textstyle\frac{1}{(x^{1}-x^{\ell})^{2}}}{+}{\textstyle\frac{1}{(x^{1}+x^{\ell})^{2}}}\bigr{\\}}}$
(94)
$\displaystyle{-2{\mathrm{i}}g_{\ell}(g_{\ell}{-}1)\textstyle\sum_{\ell\neq
1}^{3}\\{p_{1},{\textstyle\frac{1}{(x^{1}-x^{\ell})^{3}}}{+}{\textstyle\frac{1}{(x^{1}+x^{\ell})^{3}}}\\}+g_{\ell}^{2}(g_{\ell}{-}1)^{2}\textstyle\sum_{\ell\neq
1}^{3}\left({\textstyle\frac{1}{(x^{1}-x^{j})^{4}}}{+}{\textstyle\frac{1}{(x^{1}+x^{j})^{4}}}\right)}$
$\displaystyle{+4g_{\ell}(g_{\ell}{-}1)\left({\textstyle\frac{1}{(x^{1}-x^{2})^{2}}}{-}{\textstyle\frac{1}{(x^{1}+x^{2})^{2}}}\right)p_{1}p_{2}+4g_{\ell}^{2}(g_{\ell}{-}1)^{2}\left({\textstyle\frac{3}{((x^{1})^{2}-(x^{2})^{2})^{2}}}{+}4{\textstyle\frac{(x^{1})^{2}+(x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}{\textstyle\frac{(x^{1})^{2}+(x^{3})^{2}}{((x^{1})^{2}-(x^{3})^{2})^{2}}}\right)}$
$\displaystyle{+8g_{\ell}(g_{\ell}{-}1)g_{s}(g_{s}{-}1)\left({\textstyle\frac{1}{(x^{1}x^{2})^{2}}}{+}{\textstyle\frac{2}{((x^{1})^{2}-(x^{2})^{2})^{2}}}\right)+\text{cyclic}\
.}$ $\displaystyle{I_{6}}$
$\displaystyle=\tfrac{1}{3}p_{1}^{2}p_{2}^{2}p_{3}^{2}+g_{s}(g_{s}{-}1)\tfrac{1}{(x^{1})^{2}}p_{2}^{2}p_{3}^{2}-g_{\ell}(g_{\ell}{-}1)\bigl{\\{}p_{1}p_{2}p_{3}^{2},{\textstyle\frac{1}{(x^{1}-x^{2})^{2}}}{-}{\textstyle\frac{1}{(x^{1}+x^{2})^{2}}}\bigr{\\}}+g_{s}^{2}(g_{s}{-}1)^{2}\tfrac{1}{(x^{1})^{2}(x^{2})^{2}}p_{3}^{2}$
(95)
$\displaystyle+16g_{\ell}^{2}(g_{\ell}{-}1)^{2}{\textstyle\frac{(x^{1}x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{4}}}p_{3}^{2}+8g_{s}(g_{s}{-}1)g_{\ell}(g_{\ell}{-}1){\textstyle\frac{1}{((x^{1})^{2}-(x^{2})^{2})^{2}}}p_{3}^{2}+16g_{\ell}^{2}(g_{\ell}{-}1)^{2}\bigl{\\{}p_{1}p_{3},{\textstyle\frac{x^{1}x^{3}(x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}((x^{2})^{2}-(x^{3})^{2})^{2}}}\bigr{\\}}$
$\displaystyle-4g_{s}(g_{s}{-}1)g_{\ell}(g_{\ell}{-}1)\bigl{\\{}p_{1}p_{3},{\textstyle\frac{x^{1}x^{3}}{(x^{2})^{2}((x^{3})^{2}-(x^{1})^{2})^{2}}}\bigr{\\}}-g_{\ell}^{2}(g_{\ell}{-}1)^{2}{\textstyle\frac{48(x^{1}x^{2})^{2}((x^{1})^{2}+(x^{2})^{2})+160(x^{1}x^{2}x^{3})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}((x^{2})^{2}-(x^{3})^{2})^{2}((x^{3})^{2}-(x^{1})^{2})^{2}}}$
$\displaystyle+g_{s}(g_{s}{-}1)g_{\ell}^{2}(g_{\ell}{-}1)^{2}\left({\textstyle\frac{1}{(x^{1})^{2}(x^{2}-x^{3})^{4}}}{+}{\textstyle\frac{1}{(x^{1})^{2}(x^{2}+x^{3})^{4}}}{+}{\textstyle\frac{32(x^{1})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}((x^{3})^{2}-(x^{1})^{2})^{2}}}\right)$
$\displaystyle-2g_{s}(g_{s}{-}1)g_{\ell}(g_{\ell}{-}1)\bigl{(}g_{\ell}(g_{\ell}{-}1){-}4g_{s}(g_{s}{-}1)\bigr{)}{\textstyle\frac{1}{((x^{1})^{2}((x^{2})^{2}-(x^{3})^{2})^{2}}}+\tfrac{1}{3}\tfrac{g_{s}^{3}(g_{s}{-}1)^{3}}{(x^{1})^{2}(x^{2})^{2}(x^{3})^{2}}+\text{cyclic}\
.$ $\displaystyle 2{J_{4}}$
$\displaystyle=\\{p_{1}^{3},x^{1}\\}+g_{s}(g_{s}{-}1)\\{p_{1},{\textstyle\frac{1}{x^{1}}}\\}+g_{\ell}(g_{\ell}{-}1)\Bigl{(}6\textstyle\sum_{\ell\neq
1}\bigl{(}{\textstyle\frac{1}{(x^{1}-x^{\ell})^{2}}}+{\textstyle\frac{1}{(x^{1}+x^{\ell})^{2}}}\bigr{)}x^{1}p_{1}-\textstyle\sum_{\ell\neq
1}\bigl{\\{}p_{1},{\textstyle\frac{1}{x^{1}-x^{\ell}}}+{\textstyle\frac{1}{x^{1}+x^{\ell}}}\bigr{\\}}\Bigr{)}$
(96) $\displaystyle+\text{cyclic}\ ,$ $\displaystyle 6{J_{6}}$
$\displaystyle=p_{2}^{2}p_{3}^{2}\\{p_{1},x^{1}\\}{+}2{\mathrm{i}}g_{\ell}(g_{\ell}{-}1)\Bigl{(}{\textstyle\frac{(p_{1}-p_{2})}{(x^{1}-x^{2})^{3}}}+{\textstyle\frac{(p_{1}+p_{2})}{(x^{1}+x^{2})^{3}}}\Bigr{)}\\{p_{3},x^{3}\\}{+}g_{s}(g_{s}{-}1)p_{3}^{2}\Bigl{(}\\{p_{1},{\textstyle\frac{x^{1}}{(x^{2})^{2}}}\\}+\\{p_{2},{\textstyle\frac{x^{2}}{(x^{1})^{2}}}\\}\Bigr{)}$
$\displaystyle-8g_{\ell}(g_{\ell}{-}1){\textstyle\frac{(x^{1}x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}\bigr{(}{\textstyle\frac{p_{1}}{x^{1}}}+{\textstyle\frac{p_{2}}{x^{2}}}\bigr{)}p_{3}^{2}{-}16g_{\ell}(g_{\ell}{-}1)\tfrac{x^{1}x^{2}x^{3}}{((x^{1})^{2}-(x^{2})^{2})^{2}}p_{1}p_{2}p_{3}{+}4{\mathrm{i}}g_{\ell}(g_{\ell}{-}1){\textstyle\frac{(x^{1})^{2}+(x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}p_{3}^{2}$
$\displaystyle+8{\mathrm{i}}g_{\ell}(g_{\ell}{-}1){\textstyle\frac{x^{1}x^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}p_{1}p_{2}{+}g_{s}^{2}(g_{s}{-}1)^{2}{\textstyle\frac{1}{(x^{1}x^{2})^{2}}}\\{p_{3},x^{3}\\}+8g_{\ell}(g_{\ell}{-}1)(2g_{\ell}^{2}{-}2g_{\ell}{-}9){\textstyle\frac{(x^{1}x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{4}}}\\{p_{3},x^{3}\\}$
$\displaystyle-4g_{\ell}(g_{\ell}{-}1)g_{s}(g_{s}{-}1)\Bigl{\\{}p_{3},\bigr{(}{\textstyle\frac{(x^{1})^{2}}{(x^{2})^{2}((x^{3})^{2}-(x^{1})^{2})^{2}}}+{\textstyle\frac{(x^{2})^{2}}{(x^{1})^{2}((x^{3})^{2}-(x^{2})^{2})^{2}}}\bigr{)}x^{3}\Bigr{\\}}+8g_{\ell}(g_{\ell}{-}1)g_{s}(g_{s}{-}1){\textstyle\frac{1}{((x^{1})^{2}-(x^{2})^{2})^{2}}}\\{p_{3},x^{3}\\}$
$\displaystyle-12g_{\ell}(g_{\ell}{-}1){\textstyle\frac{(x^{1})^{4}+(x^{2})^{4}}{((x^{1})^{2}-(x^{2})^{2})^{4}}}\\{p_{3},x^{3}\\}+16g_{\ell}^{2}(g_{\ell}{-}1)^{2}{\textstyle\frac{(x^{1}x^{2})^{2}}{((x^{1})^{2}-(x^{2})^{2})^{2}}}\Bigl{\\{}p_{3},\bigr{(}{\textstyle\frac{1}{((x^{3})^{2}-(x^{1})^{2})^{2}}}+{\textstyle\frac{1}{((x^{2})^{2}-(x^{3})^{2})^{2}}}\bigr{)}x^{3}\Bigr{\\}}+\text{cyclic}\
.$

##

## References

  * [1] F. Calogero,
Solution of the one-dimensional N-body problem with quadratic and/or inversely
quadratic pair potentials,
J. Math. Phys. 12 (1971) 419–436; Erratum, ibidem 37 (1996) 3646.

  * [2] S. Wojciechowski,
Superintegrability of the Calogero–Moser system, Phys. Lett. 95A (1983)
279–281.

  * [3] M.A. Olshanetsky, A.M. Perelomov,
Quantum integrable systems related to Lie algebras,
Phys. Rept. 94 (1983) 313–404.

  * [4] A.P. Polychronakos,
Physics and mathematics of Calogero particles,
J. Phys. A: Math. Gen. 39 (2006) 12793, arXiv:hep-th/0607033.

  * [5] A.P. Polychronakos,
Exchange interactions, Yang–Baxter relations and transparent particles,
Nucl. Phys. B 961 (2020) 115243, arXiv:2006.14624 [hep-th].

  * [6] V.B. Kuznetsov,
Hidden symmetry of the quantum Calogero–Moser system,
Phys. Lett. A 218 (1996) 212–222, arXiv:solv-int/9509001 [nlin.SI].

  * [7] F. Correa, O. Lechtenfeld, M. Plyushchay,
Nonlinear supersymmetry in the quantum Calogero model,
JHEP 04 (2014) 151, arXiv:1312.5749 [hep-th].

  * [8] C.F. Dunkl,
Differential difference operators associated to reflection groups,
Trans. Am. Math. Soc. 311 (1989) 167–183.

  * [9] M. Rösler,
Dunkl operators: theory and applications,
lecture notes for the SIAM activity group OP-SF summer school 2002; Leuven,
Belgium.

  * [10] F. Correa and O. Lechtenfeld,
The tetrahexahedric angular Calogero model,
JHEP 10 (2015) 191, arXiv:1508.04925 [hep-th].

  * [11] F. Correa and O. Lechtenfeld,
$\mathcal{P}\mathcal{T}$ deformation of angular Calogero models,
JHEP 11 (2017) 122, arXiv:1705.05425 [hep-th].

  * [12] F. Correa and O. Lechtenfeld,
$\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models,
JHEP 05 (2019) 166 arXiv:1903.06481 [hep-th].


# Optimising Resource Management for
Embedded Machine Learning

Lei Xun, Long Tran-Thanh, Bashir M Al-Hashimi, Geoff V. Merrett  School of
Electronics and Computer Science
University of Southampton
Southampton, UK
{lx2u16, ltt08r, bmah<EMAIL_ADDRESS>
###### Abstract

Machine learning inference is increasingly being executed locally on mobile
and embedded platforms, due to the clear advantages in latency, privacy and
connectivity. In this paper, we present approaches for online resource
management in heterogeneous multi-core systems and show how they can be
applied to optimise the performance of machine learning workloads. Performance
can be defined using platform-dependent (e.g. speed, energy) and platform-
independent (accuracy, confidence) metrics. In particular, we show how a Deep
Neural Network (DNN) can be dynamically scalable to trade-off these various
performance metrics. Achieving consistent performance when executing on
different platforms is necessary yet challenging, due to the different
resources provided and their capability, and their time-varying availability
when executing alongside other workloads. Managing the interface between
available hardware resources (often numerous and heterogeneous in nature),
software requirements, and user experience is increasingly complex.

###### Index Terms:

Embedded Machine Learning, Dynamic Deep Neural Network, Runtime Resource
Management

##  I Introduction

Deep Neural Networks (DNNs)[1] are widely used in many applications including
computer vision [2, 3] and natural language processing[4]. Compared to
traditional hand-engineered machine learning algorithms, DNNs have
demonstrated near or super-human accuracy.

Figure 1: DNNs can be deployed on a variety of hardware platforms with
different computing resources. At design time, the DNN is compressed using
techniques such as static model pruning [5, 6] (see Section III), and then
mapped onto different computing resources to meet the performance requirements
of the application.

The execution of DNNs has two stages: training and inference. At the training
stage, DNNs learn the rules to execute specific tasks from a corresponding
dataset. During this process, millions of DNN parameters are adjusted, hence
training is usually executed on cloud GPU(s). At the inference stage, the DNN
is loaded with pre-trained parameters to execute the tasks, and the parameters
are not changed. Inference can be executed on cloud GPU(s), where the data is
sent from end-users to the cloud for processing and the inference result is
returned. However, there is increasing interest in moving inference to be
executed locally on mobile and embedded platforms due to a number of
advantages. First, executing DNNs on the device can offer lower latency than
executing them in the cloud. This lower latency leads to improved user
experience, and/or the ability to meet the strict timing requirements of real-
time applications such as self-driving cars [7]. Second, mobile and embedded
devices often collect and process personal data; keeping these data locally
helps to mitigate potential privacy concerns [8]. Finally, in areas with slow,
intermittent or non-existent internet connectivity, timely cloud access can be
difficult. Performing inference locally allows this issue to be mitigated.

It is attractive to be able to deploy DNN inference on a variety of platforms
with distinct different computing resources, and to meet diverse application
requirements (Fig 1). However, the high accuracy of DNNs comes at the cost of
high computational requirements. DNNs are often too computationally intensive
for resource-constrained platforms like mobile and embedded platforms[5, 9],
and this makes their efficient execution demanding and challenging. To
efficiently execute DNNs on mobile and embedded platforms, a significant
amount of recent work has focused on specialist hardware accelerator (also
known as a Neural Processing Unit (NPU)) design [10, 11, 12]. Design-
time/offline approaches such as static model pruning[5, 6] have been proposed
to compress DNN models according to the application requirements and target
hardware. However, while this can be applied offline to compress the model to
approximately the ‘right size’, managing the interface between available
hardware resources (often numerous and heterogeneous in nature), software
requirements, and user experience all of which are typically intractable at
design-time. At runtime, since modern System on Chips (SoCs) typically execute
a combination of different and dynamic workloads concurrently, it is
challenging to consistently meet inference time/energy budgets because the
local computing resources available to the DNN vary considerably (Fig 2). A
variety of dynamic DNNs have been proposed to dynamically change the number of
active filters to trade-off accuracy for time/energy reduction [9, 13].
However, these approaches did not explore optimisation opportunities in
hardware (e.g. Dynamic Voltage and Frequency Scaling (DVFS) and task mapping).

In this paper, we motivate the opportunities in online resource management for
DNN inference on mobile and embedded platforms (Section II), and explore how
state-of-the-art approaches are enabling the dynamic performance scaling of
DNNs that can be applied (Section III). In particular, we use a case study
(Section IV) to show how a Deep Neural Network (DNN) can be dynamically
scalable to trade-off these various performance metrics. We then identify the
opportunities for runtime resource management in this area, such that the
performance trade-offs in both application and device can be explored and
managed (Section V).

##  II Motivation for Online Resource Management

Figure 2: At runtime, the local computing resources available to DNNs may vary
considerably due to multiple applications running concurrently.

Modern mobile and embedded SoCs typically contains multiple heterogeneous
computing cores, such as CPUs, GPUs, DSPs, FPGAs, etc. In the last few years,
the inclusion of NPU has become a trend to improve the efficiency of DNN
inference execution. For example, the Huawei Kirin 990 5G [14] contains 8 CPU
cores of three different types, a 16-core GPU and an NPU with 3 cores of two
different types. The Apple A13 Bionic[15] contains 6 CPU cores of two
different types, a quad-core GPU and an NPU with 8 cores. In addition,
dedicated matrix multipliers are attached to the CPU cluster for the
acceleration of machine learning workloads.

It is attractive to be able to deploy DNN applications on a variety of
different hardware platforms while consistently meeting pre-defined
performance requirements. The performance of inference can be defined using
platform-dependent (e.g. speed, energy) and platform-independent (accuracy,
confidence) metrics. As shown in Table I, when the same DNN model is deployed
across different hardware platforms/cores, the execution time, energy and
power consumption of inference vary considerably, but the accuracy remains the
same.

Two main challenges exist at design time and runtime. Different hardware
platforms have vastly different computing resources and capabilities, and
different applications have different performance requirements. Approaches are
therefore needed to adapt DNN models to different hardware accordingly. This
can be solved, at design-time (Fig 1), by ‘compressing’ the DNN for target
platform. For example, the same DNN might be used uncompressed on one platform
with an NPU, while a compressed model (with offering lower accuracy but
requiring fewer computations) is deployed on a different platform containing
only CPU and GPU cores in order to meet the same execution time and energy
consumption requirements.

TABLE I: Platform-dependent & independent DNN performance metrics Platform | Computing cores | Platform-dependent metrics | Platform-independent metrics
---|---|---|---
Execution time (ms) | Power (mW) | Energy (mJ) | Top-1 Accuracy (%)
Jetson Nano | GPU (614MHz) + A57 CPU (921MHz) | 7.4 | 1340 | 9.92 | 71.2
GPU (921MHz) + A57 CPU (1.43GHz) | 4.93 | 2500 | 12.3
A57 CPU (921MHz) | 69.4 | 878 | 60.9
A57 CPU (1.43GHz) | 46.9 | 1490 | 69.9
Odroid XU3 | A15 CPU (200MHz) | 1020 | 326 | 320
A15 CPU (1GHz) | 204 | 846 | 173
A15 CPU (1.8GHz) | 117 | 2120 | 248
A7 CPU (200MHz) | 1780 | 72.4 | 129
A7 CPU (700MHz) | 504 | 141 | 71.4
A7 CPU (1.3GHz) | 280 | 329 | 92.1

While design-time approaches compress a model to a size that should be
executable on the target platform, at runtime, the available computing
resources may vary considerably due to multiple applications executing
concurrently. Fig 2 shows an example of how available computing resources may
vary at runtime when different applications are executing:

  * •

Single DNN [t=0s, Fig 2(a)]: at the beginning, when there is only one DNN
running on the platform, the NPU is used along with a CPU for pre-processing
(e.g. image resizing). This may be the ‘compressed’ model that was created for
the platform at design time.

  * •

Two DNNs [t=5s, Fig 2(b)]: a second DNN is deployed on the platform. It has
higher requirements on the desired classification execution time, therefore it
is executed on the NPU, and the first DNN is migrated to the GPU. Since GPU is
typically slower than NPU for machine learning workloads, the first DNN is
dynamically compressed which requires fewer computations by trading accuracy
(see Section III). This makes sure the performance requirements of two DNNs
are both met.

  * •

Two DNNs and a VR/AR application [t=15s, Fig 2(c)]: a VR/AR application is
deployed onto the GPU, therefore the first DNN is migrated to the big CPU
cluster and all four CPU cores are used due to the sheer volume of
computations. Shortly after, the temperature of the SoC exceeds thermal
limits. Therefore, the first DNN is dynamically compressed further and mapped
onto a single core CPU in order to meet system thermal budgets.

  * •

Application performance requirement is changed [t=25s, Fig 2(d)]: the accuracy
requirement of the second DNN is reduced (e.g. by the user), therefore it can
be dynamically compressed to a smaller model configuration and offers the
spare NPU memory and computing capabilities to the first DNN. Two DNNs are
dynamically scaled together to a model configuration where they are deployed
on the same NPU.

Existing approaches for efficient execution of DNNs mainly focus on either
hardware or software approaches. Although these works expose the optimisation
opportunities on both sides, managing the interface between available hardware
resources, software requirements, and user experience is not addressed, this
identifies the opportunities for runtime resource management in this area.

##  III Approaches for Efficient Execution of DNNs

This section introduces state-of-the-art works on the efficient execution of
DNNs from both the hardware and software perspectives. In particular, hardware
accelerators for machine learning and static model pruning are widely used to
address design time challenges, and dynamic model pruning approaches are used
to let DNN consistently meet the performance requirements while running on
dynamically available computing resources.

###  III-A Specialist DNN hardware accelerator

A significant amount of work has been proposed from both academia (e.g.
DianNao [10], EIE[12] and Eyeriss v2 [11]) and industry (e.g. Apple A13
Bionic[15] and Huawei Kirin 990 5G[14]).

In a typical DNN, the most computational intensive operation is matrix
multiplication. Therefore, NPUs usually contain dedicated matrix multipliers,
and since data movement dominates the energy consumption in DNN
computation[16], large on-chip memories are used to reduce off-chip accesses.
It is attractive to be able to execute all DNNs on NPU, but not every hardware
platform contains an NPU, and multiple DNNs executing concurrently will
compete for the limited NPU resources at runtime. Mapping DNNs onto CPUs or
GPUs may result in the violation of inference execution time and energy
budgets. Therefore, DNN model pruning is also needed.

Figure 3: Dynamic DNN using incremental training and group convolution
pruning. The (a) channels of the DNN convolution layer are divided into
groups, which are then trained incrementally (b). At runtime, (c) later groups
can be pruned for inference time/energy reduction or added back for accuracy
recovery without model retraining.

###  III-B Static Model Pruning

Weight pruning is a static model compression approach to reduce the number of
parameters in a DNN, Han et al. [17] proposed a magnitude-based algorithm,
where parameters with small values are pruned. However, this approach leads to
unstructured sparse filters which cannot be accelerated by most hardware[18],
except hardware accelerators that are designed specifically for sparse
DNNs[12, 11]. Unlike weight pruning, filter pruning[19] prunes whole filters
to compress the model. Although this approach has a lower model compression
rate compared to weight pruning, it does not generate unstructured filters,
and hence can gain actual acceleration on all hardware platforms.

Yang et al. [5] use filter pruning to compress DNNs to a meet pre-defined
inference execution time budget on any target hardware platform. This approach
offers a trade-off between accuracy and execution time. To achieve consistent
performance across different platforms, the DNN is compressed more on
platforms with less computing capabilities (i.e. sacrificing more accuracy for
a reduction in execution time). For example, the full DNN model can be
deployed on an NPU, yet the smaller compressed models can be deployed on GPUs
and CPUs to meet the same time budget with less accuracy. This approach
generates one DNN for a given performance budget at a pre-defined hardware
setting (e.g. computing core and voltage/frequency level).

This raises a significant problem, since modern SoCs typically execute a
combination of different dynamic workloads concurrently, and hence the local
resources available to the DNN vary considerably at runtime. The performance
budgets cannot be met when the pre-defined hardware setting is unavailable at
runtime. For example, the computing core may be unavailable because other
applications are running on it, or available at a lower voltage/frequency due
to other computing cores executing in the same voltage/frequency domain, or
fixed power/thermal budgets. Multiple DNNs are needed to cover all hardware
settings, which result in significant memory storage overhead. Furthermore,
the switching activities of these DNNs at runtime may cause significant delay
and energy consumption[20].

###  III-C Dynamic Model Pruning

Static model pruning generates one DNN for a given performance budget and
hardware setting combination. Multiple DNNs need to be generated to cover all
hardware combinations (i.e. core, voltage and frequency) in an SoC. Dynamic
Model Pruning (also known as dynamic DNNs) contains multiple DNN
configurations in a single model. These configurations use a different number
of filters within the same model, hence they are stored within a single model
memory footprint and have different sizes, accuracies and computation
requirements.

Dynamic DNNs [21, 9, 13, 22] can be partially executed to consistently meet
the performance budget (e.g. time, energy), while adapting to runtime resource
varieties on the hardware. For example, smaller DNN configurations that are
less accurate but require less computation are deployed when the computing
capabilities of the hardware resources available at runtime are reduced (e.g.
at a lower voltage/frequency).

##  IV Case Study: Exploring Performance Trade-offs with Dynamic DNNs

Modern mobile and embedded SoCs are highly efficient because of the use of
runtime resource management techniques, such as scheduling task mapping, DPM
and DVFS. Although the dynamic DNNs presented above are scalable to trade-off
various performance metrics, they can be combined with task mapping and DVFS
to achieve a wider dynamic range of performance trade-off. To illustrate this,
we proposed a dynamic DNN that can be scaled with task mapping and DVFS[23].
The DNN is built using incremental training and group convolution pruning, and
is shown in Fig 3. The channels of the DNN’s convolution layers are divided
into groups (Fig 3 (a)), which are then trained incrementally (Fig 3 (b)). At
runtime, later groups can be pruned for inference time/energy reduction, or
added back for accuracy recovery when more computing resources become
available. This is all possible at runtime without model retraining (Fig 3
(c)).

We use a four-increment design to obtain four different DNN configurations,
and refer them as the 25%, 50%, 75% and 100% models. The impact on
classification accuracy is shown in Fig 4(b). The 25% model uses only one
group of DNN parameters; therefore it is the least accurate model but requires
the minimum computation. The 100% model is the full model; therefore it is the
most accurate and computationally expensive model. The design is validated
through empirical measurements on the Odroid XU3 heterogeneous multi-core
platform. The DNN is mapped on both Arm A15 and A7 CPU cores, and under 17 and
12 different frequency levels respectively. The results are shown in Fig 4(a).
Task mapping, DVFS and the dynamic DNN can be seen as three adjustable knobs
which can be adjusted to meet dynamic E, P, t and accuracy budgets/targets at
runtime. For example, in Fig 4, for a budget of 400 ms and 100 mJ, a 100%
model on the A7 CPU at 900 MHz could offer the highest accuracy and lowest
energy consumption. If the budgets change to 200 ms and 150 mJ, then a 75%
model on the A15 CPU at 1 GHz becomes the new optimal configuration. In the
case of multiple applications executing concurrently, the frequency setting
may be sub-optimal due to other applications in the same frequency domain are
using that frequency level.

Figure 4: (a) E, t operating points sapce. A Dynamic DNN (different colours
show different levels of ‘compression’) is combined with task mapping
(different symbols) and DVFS (different points) on the Odroid XU3 platform.
(b) Top-1 image classification accuracy on 10,000 CIFAR10 validation images.
The error bar shows the variance over 10 image classes of CIFAR10.

##  V Online Resource Management of Machine Learning

The above approaches offer a number of dynamically selectable operating point
in the E, P, t, accuracy space. However, to meet the opportunities of Fig 2,
these approaches need to be integrated with online resource management.

A variety of online resource management approaches have been proposed, such as
DVFS [24], task mapping [25] and power gating [26]. These approaches optimise
hardware behaviour to satisfy constraints (e.g. temperature, power); the
performance requirements and optimisation opportunities in the application are
traditionally not addressed.

Runtime management (RTM) can be enhanced by using ‘knobs’ and ‘monitors’ of
the application and device, which provide interfaces to convey information
between RTM, applications and devices. A variety of works have been proposed
[27, 28, 29, 30], which focus on the opportunities in either applications or
devices. It is necessary to explore the opportunities on both sides at the
same time to address the sheer volume of computation in DNN applications.
Bragg et al. [31] proposed the PRiME framework, that abstracts the system into
three layers: application, device and RTM; control between them operates
through knobs and monitors in the application and hardware devices. In
particular, knobs are adjustable parameters in the application (e.g. execution
iteration, data precision) and device (e.g. voltage/frequency, core), and
monitors offer performance information about the application (e.g. accuracy,
frame rate) and device (e.g. power, temperature).

Fig 5 shows how dynamic DNN, task mapping and DVFS can be combined together
alongside an RTM framework such as PRiME. The opportunities for DNN
performance trade-off in both hardware and software are managed through knobs
and monitors that are controlled by RTM. For example, dynamic DNN (application
knob) can be scaled with DVFS and task mapping (device knobs) to meet the DNN
performance requirements (application monitors), and meet any hardware
temperature and power constraints (device monitors). In the case of Fig 2 (d),
the RTM received new accuracy requirements for the DNNs through the
application monitor, and hence RTM changed the size of the DNNs using the
application knob, mapping them onto the NPU using the device knob. In
addition, system physical limits like temperature and power are monitored
using the device monitors, as DVFS could be then applied in order to meet
these limits.

Figure 5: Apply state-of-the-art runtime resource management for embedded
machine learning. The DNN performance trade-off opportunities in the hardware
and software are managed through knobs and monitors controlled by RTM.

##  VI Conclusions

This paper has presented the challenges of deploying DNNs on mobile and
embedded platforms, at both design time and runtime. At design time, DNNs are
deployed on a variety of hardware platforms with vastly different computing
resources to meet different application requirements. However, at runtime, it
is very challenging to consistently meet performance requirements, since the
availability of the local computing resources vary considerably due to other
applications executing concurrently. We showed how approaches enabling the
dynamic performance scaling of DNNs can be applied to address these
challenges, and proposed how online resource management approaches can be
applied to manage and optimise machine learning workloads alongside other
applications at runtime. Execution of DNN inference on mobile and embedded
platforms is clearly important, but also challenging, and we believe that
further research is necessitated in runtime resource allocation and adaptation
to optimise this.

##  VII Acknowledgement

This work was supported in part by the Engineering and Physical Sciences
Research Council (EPSRC) under Grant EP/S030069/1. Experimental data can be
found at DOI: 10.5258/SOTON/D1154

## References

  * [1] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” _Nature_ , vol. 521, no. 7553, p. 436, 2015.
  * [2] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in _Advances in Neural Information Processing Systems (NeurIPS)_ , 2012, pp. 1097–1105.
  * [3] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, “You only look once: Unified, real-time object detection,” in _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016, pp. 779–788.
  * [4] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” in _Advances in Neural Information Processing Systems (NeurIPS)_ , 2017, pp. 5998–6008.
  * [5] T.-J. Yang _et al._ , “Netadapt: Platform-aware neural network adaptation for mobile applications,” in _European Conference on Computer Vision (ECCV)_ , 2018, pp. 285–300.
  * [6] Y. He, J. Lin, Z. Liu, H. Wang, L.-J. Li, and S. Han, “Amc: Automl for model compression and acceleration on mobile devices,” in _European Conference on Computer Vision (ECCV)_ , 2018, pp. 784–800.
  * [7] V. Sze, Y.-H. Chen, T.-J. Yang, and J. S. Emer, “Efficient processing of deep neural networks: A tutorial and survey,” _Proceedings of the IEEE_ , vol. 105, no. 12, pp. 2295–2329, 2017.
  * [8] D. Lynskey, “Alexa, are you invading my privacy?” [Online]. Available: https://www.theguardian.com/technology/2019/oct/09/alexa-are-you-invading-my-privacy-the-dark-side-of-our-voice-assistants.
  * [9] Z. Xu, F. Yu, C. Liu, and X. Chen, “Reform: Static and dynamic resource-aware dnn reconfiguration framework for mobile device,” in _Design Automation Conference (DAC)_ , 2019, p. 183.
  * [10] T. Chen, Z. Du, N. Sun, J. Wang, C. Wu, Y. Chen, and O. Temam, “Diannao: A small-footprint high-throughput accelerator for ubiquitous machine-learning,” in _ASPLOS_ , 2014, pp. 269–284.
  * [11] Y.-H. Chen, T.-J. Yang, J. Emer, and V. Sze, “Eyeriss v2: A flexible accelerator for emerging deep neural networks on mobile devices,” _IEEE Journal on Emerging and Selected Topics in Circuits and Systems_ , 2019.
  * [12] S. Han, X. Liu, H. Mao, J. Pu, A. Pedram, M. A. Horowitz, and W. J. Dally, “EIE: efficient inference engine on compressed deep neural network,” in _International Symposium on Computer Architecture (ISCA)_ , 2016, pp. 243–254.
  * [13] H. Tann, S. Hashemi, R. Bahar, and S. Reda, “Runtime configurable deep neural networks for energy-accuracy trade-off,” in _International Conference on Hardware/Software Codesign and System Synthesis_. ACM, 2016, p. 34.
  * [14] “Huawei kirin 990 series,” [Online]. Available: https://consumer.huawei.com/en/campaign/kirin-990-series/.
  * [15] J. Cross, “Inside Apple’s A13 Bionic system-on-chip,” [Online]. Available: https://www.macworld.com/article/3442716/inside-apples-a13-bionic-system-on-chip.html.
  * [16] Y.-H. Chen, J. Emer, and V. Sze, “Eyeriss: A spatial architecture for energy-efficient dataflow for convolutional neural networks,” in _International Symposium on Computer Architecture (ISCA)_ , 2016, pp. 367–379.
  * [17] S. Han, J. Pool, J. Tran, and W. Dally, “Learning both weights and connections for efficient neural network,” in _Advances in Neural Information Processing Systems (NeurIPS)_ , 2015, pp. 1135–1143.
  * [18] J. Yu, A. Lukefahr, D. Palframan, G. Dasika, R. Das, and S. Mahlke, “Scalpel: Customizing dnn pruning to the underlying hardware parallelism,” in _International Symposium on Computer Architecture (ISCA)_ , 2017, pp. 548–560.
  * [19] H. Li, A. Kadav, I. Durdanovic, H. Samet, and H. P. Graf, “Pruning filters for efficient convnets,” _arXiv preprint arXiv:1608.08710_ , 2016.
  * [20] E. Park, D. Kim, S. Kim, Y.-D. Kim, G. Kim, S. Yoon, and S. Yoo, “Big/little deep neural network for ultra low power inference,” in _International Conference on Hardware/Software Codesign and System Synthesis_. IEEE Press, 2015, pp. 124–132.
  * [21] J. Lin, Y. Rao, J. Lu, and J. Zhou, “Runtime neural pruning,” in _Advances in Neural Information Processing Systems (NeurIPS)_ , 2017, pp. 2181–2191.
  * [22] B. Fang, X. Zeng, and M. Zhang, “Nestdnn: Resource-aware multi-tenant on-device deep learning for continuous mobile vision,” in _International Conference on Mobile Computing and Networking_. ACM, 2018, pp. 115–127.
  * [23] L. Xun, L. Tran-Thanh, B. M. Al-Hashimi, and G. V. Merrett, “Incremental training and group convolution pruning for runtime dnn performance scaling on heterogeneous embedded platforms,” in _Workshop on Machine Learning for CAD (MLCAD)_ , 2019.
  * [24] A. Das, R. A. Shafik, G. V. Merrett, B. M. Al-Hashimi, A. Kumar, and B. Veeravalli, “Reinforcement learning-based inter-and intra-application thermal optimization for lifetime improvement of multicore systems,” in _Design Automation Conference (DAC)_ , 2014, pp. 1–6.
  * [25] B. K. Reddy, A. K. Singh, D. Biswas, G. V. Merrett, and B. M. Al-Hashimi, “Inter-cluster thread-to-core mapping and dvfs on heterogeneous multi-cores,” _Transactions on Multi-Scale Computing Systems_ , vol. 4, no. 3, pp. 369–382, 2017.
  * [26] A. M. Rahmani, M.-H. Haghbayan, A. Miele, P. Liljeberg, A. Jantsch, and H. Tenhunen, “Reliability-aware runtime power management for many-core systems in the dark silicon era,” _IEEE Transactions on Very Large Scale Integration (VLSI) Systems_ , vol. 25, no. 2, pp. 427–440, 2016.
  * [27] B. Donyanavard, T. Mück, A. M. Rahmani, N. Dutt, A. Sadighi, F. Maurer, and A. Herkersdorf, “Sosa: Self-optimizing learning with self-adaptive control for hierarchical system-on-chip management,” in _International Symposium on Microarchitecture (MICRO)_. ACM, 2019, pp. 685–698.
  * [28] K. Moazzemi, B. Maity, S. Yi, A. M. Rahmani, and N. Dutt, “Hessle-free: Heterogeneous systems leveraging fuzzy control for runtime resource management,” _ACM Transactions on Embedded Computing Systems (TECS)_ , vol. 18, no. 5s, p. 74, 2019.
  * [29] D. Gadioli, G. Palermo, and C. Silvano, “Application autotuning to support runtime adaptivity in multicore architectures,” in _International Conference on Embedded Computer Systems: Architectures, Modeling, and Simulation (SAMOS)_. IEEE, 2015, pp. 173–180.
  * [30] S. T. Fleming and D. B. Thomas, “Heterogeneous heartbeats: A framework for dynamic management of autonomous socs,” in _International Conference on Field Programmable Logic and Applications (FPL)_. IEEE, 2014, pp. 1–6.
  * [31] G. M. Bragg, C. R. Leech, D. Balsamo, J. J. Davis, E. Weber Wachter, G. Merrett, G. A. Constantinides, and B. Al-Hashimi, “An application-and platform-agnostic control and monitoring framework for multicore systems,” in _International Conference on Pervasive and Embedded Computing (PEC)_ , 2018\.