Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group $SO(3)$. \footnote{AMS Subject Classification 58F07, 70H99, 76B15}
Yuri N. Fedorov
Department of Mathematics and Mechanics
Moscow
119 899
Russia
and
Department de Matem\`atica I
Universitat Politecnica de Catalunya
Barcelona
E-08028 Spain
We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map $\cal B$ on the 4-dimensional variety ${\cal V}(2,3)$. The map admits two different reductions, namely, to the Lie group $SO(3)$ and to the coalgebra $so^*(3)$. The first reduction provides a discretization of the motion of the classical Euler top in space and has a transparent geometric interpretation, which can be regarded as a discrete version of the celebrated Poinsot model of motion and which inherits some properties of another discrete system, the elliptic billiard. The reduction of $\cal B$ to $so^*(3)$ gives a new explicit discretization of the Euler top in the angular momentum space, which preserves first integrals of the continuous system.
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\begin{document}

\title{Integrable flows and Backlund transformations on extended Stiefel varieties 
with application to the Euler top on the Lie group $SO(3)$. 
\footnote{AMS Subject Classification 58F07, 70H99, 76B15} }

\author{Yuri N. Fedorov \\
Department of Mathematics and Mechanics 
\\ Moscow Lomonosov University, Moscow, 119 899, Russia \\ 
e-mail: fedorov@mech.math.msu.su \\ 
and \\ 
 Department de Matem\`atica I, \\ 
Universitat Politecnica de Catalunya, \\
Barcelona, E-08028 Spain \\
e-mail: Yuri.Fedorov@upc.es  }
\maketitle

{\it Short title}: Flows and Backlund transformations on extended Stiefel varieties 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract} 
We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented
as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended 
Stiefel variety $\bar{\cal V}(k,m)$, 
and present its Lax representation with a rational parameter.

We also describe an integrable two-valued symplectic map  $\cal B$ on the 4-dimensional variety
${\cal V}(2,3)$. 
The map admits two different reductions, namely, to the Lie group $SO(3)$ and to the coalgebra $so^*(3)$.

The first reduction provides a discretization of the motion of the classical Euler top in space
and has a transparent geometric interpretation, which can be regarded as a discrete version of the
celebrated Poinsot model of motion and which inherits some properties of another discrete system, 
the elliptic billiard.  

The reduction of  $\cal B$ to $so^*(3)$ gives a new explicit discretization  of the Euler top
in the angular momentum space, which preserves first integrals of the continuous system. 
\end{abstract}

\section{Introduction}
In most publications the integrable $m$-dimensional Euler top is represented as a flow on  
the cotangent bundle $T^* SO(m)$ or on the coalgebra $so^*(m)$.

Recently, an alternative description of this problem as a system on a 
symplectic subvariety of the group product 
$SO(m) \times SO(m)$ was  proposed in \cite{Bl_Cr, BCMR}. 
 
A first discretization of the free $m$-dimensional top on $T^* SO(m)$ was 
constructed in \cite{Ves, MosVes}
by the method of factorization of matrix polynomials. 
This discretization is represented by a second order Lagrangian correspondence,
which does not explicitly involve a time step, it is determined by initial data 
(a choice of two subsequent points on $SO(m)$). 

On the other hand, in \cite{S} (see also \cite{SB}) 
Suris introduced a concept of an integrable discretisation of a 
finite-dimensional Hamiltonian system as a one parameter family of integrable Poisson maps 
parameterized by a time step  $\epsilon$, which 
differ from the identity map by $O(\epsilon)$, and whose Poisson structure and the  integrals of motion 
differ at most by  $O(\epsilon)$ from those of the continuous-time system. 

In the special case where the discretization preserves exactly both the Poisson structure 
and the integrals of motion, one speak of an ``exact discretization'': 
one has a family of B\"acklund transformations, which map solutions into solutions 
and are interpolated by a hamiltonian flow generated by some function of the integrals of 
motion of the continuous system. 

A class of implicitly defined Poisson maps $so^*(3) \to so^*(3)$ discretizing the classical Euler top 
in the space of the angular momentum  was indicated in \cite{Bob_Lorb}. 
The maps preserve the energy and momentum integrals  of  the continuous problem and contain explicitly 
a time step parameter.
It was shown that such a map preserves the standard Lie--Poisson structure on $so^*(3)$  
if and only if its restriction onto complex invariant manifolds, open subsets of elliptic curves,
is a shift, which is constant on each curve. 

Recently, another integrable discretization of the top on $so^*(3)$, which is {\it explicit}, but does not preserve 
the integrals of the continuous problem was found in \cite{Jap} by applying the Hirota method.
\medskip

\paragraph{Contents of the paper.} Our aim is twofold. First, in Section 2,
we propose yet another description of the continuous $n$-dimensional Euler--Manakov top as a reduction of
a Hamiltonian system on so called extended Stiefel variety $\bar{\cal V}(k,m)$,  a symplectic submanifold 
of dimension $km-k^2/2$ in ${\mathbb R}^{km}$, where $2\le k\le n$ is an even integer. 
We present a Lax representation of this system with a rational parameter, which, in a sense described
below, is dual to Manakov's Lax pair found in \cite{Man_so_n}.
 
The system possesses $k/2$ commuting symmetry fields  ${\cal R}_l$ generated by Hamiltonians $H_l$. 
Its  Marsden--Weinstein reduction with respect to the action of  the fields gives rise to 
a Hamiltonian system on a rank $k$ coadjoint orbit ${\cal S}_h^{(k)}$ in the coalgebra $so^*(m)$, 
whereas the original Poisson structure in $\bar{\cal V}(k,m)$ is a pull-back of 
 the standard Lie--Poisson structure of  $so^*(m)$ restricted onto the orbit. 
The reduced Hamiltonian system coincides with the  Euler--Manakov system on
${\cal S}_h^{(k)}$. In case of the maximal rank $k$, the level variety $\{H_l=c_l \}\subset \bar{\cal V}(k,m)$ 
is the group $SO(m)$, and the restriction of the original system onto the group yields a flow describing
the motion of the $n$-dimensional top in space. 

Second, in Section 3, we present an intertwining relation (discrete Lax pair) generating 
a explicit $\lambda^*$-depended family of two-valued complex B\"acklund 
transformations  $\cal B_\lambda^*$ of the variety ${\cal V}(2,3)$, 
which preserve the above Poisson structure and the first integrals of the continuous 
Hamiltonian system (formula (\ref{4.10})).  

The restricton of  $\cal B_\lambda^*$ onto the group $SO(3)$ 
provides a discretization of the motion of the classical Euler top in space and
has a transparent geometric interpretation, which, in turn, 
can be regarded as a discrete version of the
celebrated Poinsot model of motion and which inherits some properties of another discrete integrable system, 
the elliptic billiard (Figure 1).  
%The restriction of  $\cal B_\lambda^*$ onto its generic invariant manifolds, 
%generalized Jacobians, is represented by a shift depending on $\lambda^*$.  

On the other hand, the reduction of  $\cal B_\lambda^*$ onto the coalgebra $so^*(3)$ gives a new explicit discretization  
of the classical Euler top, which also preserves its first integrals (formula (\ref{diff})). 

%Finally, by modifying the transformation  $\cal B_\lambda^*$, we obtain  
%a discretization of the motion of the classical Euler top in space. 
Like the Moser--Veselov correspondence,
the both discretizations do not explicitly involve a time step and their continuous limits depend on  
the parameter $\lambda^*$.

\section{Hamiltonian Systems on Extended Stiefel Varieties 
and Rank $k$ Solutions of Frahm--Manakov top}

Recall that the free motion of an $m$-dimensional rigid body is described by 
the Euler--Frahm equations (\cite{FFrahm}) 
\begin{equation} 
\dot M=[M,\Omega ],  
\label{2.1}
\end{equation}
where $\Omega \in so(m)$ is the angular velocity,
$M\in so^*(m)$ the angular momentum of the body in the moving frame. 
Following \cite{MiFo, R}, these equations  are Hamiltonian 
with respect to  the degenerate Lie--Poisson bracket on $so^*(m)$ 
\begin{equation} \label{poinsot}
\{  M_{ij},  M_{kl} \}_{so(m)} = \delta_{il} M_{jk} - \delta_{il} M_{kj}
+\delta_{kj} M_{jl} - \delta_{ik} M_{jl}  
\end{equation}
and $\Omega_{ij}=\partial H(M)/\partial M_{ij}$. 

The restriction of $\{\cdot, \cdot  \}_{so(m)}$ onto orbits of coadjoint action of
$SO(m)$  in $so^*(m)$ is nondegenerate.  A generic orbit ${\cal S}_h$ parameterized by 
$[m/2]$ independent Casimir functions of the bracket is thus a symplectic variety of dimension
$m(m-1)/2-[m/2]$. 

% have theform
%\Omega ={\partial H(M)\over \partial M}, $H(M)$ is the Hamiltonian, 

Equations (\ref{2.1}) are known to
be integrable provided $M$ and $\Omega$ are related as
$[M, a]=[\Omega, b]$, where $a,b$ are constant commuting matrices, 
and all the eigenvalues of $a$ and $b$ are distinct.
The integrability follows from the  Lax representation
with a rational spectral parameter found by Manakov in \cite{Man_so_n}, 
or from 
a hyperelliptic Lax pair indicated in \cite{Fe_AMS}. These Lax pairs provide a complete set
of integrals of motion, whose involutivity  can be proved by applying $r$-matrix theory. 


For the concreteness, 
in the sequel we consider the case $a=\mbox{diag}(a_1,\dots,a_m)$, $b=a^2$.
Then $\Omega=AM+M A$, and equations (\ref{2.1}) take the form
\begin{equation}  
\label{FM_again}
\dot M=[M,aM+Ma]
\end{equation}

 Apart from this ``basic'' system, there exists a whole hierarchy of
``higher Manakov systems'',  which are defined by different relations between $\Omega$ and $M$,
and which commute with (\ref{FM_again}).
\medskip

Below we show that the restrictions of the Frahm--Manakov system on rank $k$ orbits of coadjoint representation 
of $SO(m)$ in  $so^*(m)$ are closely related to certain Hamiltonian 
dynamical systems on extended Stiefel varieties. 
Recall that the standard Stiefel variety ${\cal V}(k,m)$ is
the variety of ordered sets of $k$ orthogonal vectors in ${\mathbb R}^m$ (${\mathbb C}^m$) 
having fixed squares. It is a smooth variety of dimension $km-k(k+1)/2$ (see e.g., \cite{Sovr_Geom}).

Namely, as follows from (\ref{2.1}), the angular momentum in space is a constant matrix.
Hence, due to the Darboux theorem, in the case rank $M=k$ there exist $k$ mutually orthogonal and
{\it fixed in space\/} vectors $x^{(l)}, y^{(l)}\in {\mathbb R}^m$, $l=1,\dots,k/2$ such that
$|x^{(l)}|^2=|y^{(l)}|^2=h_l$ and the momentum $M$ can be represented in form
\begin{gather}
M=\sum_{l=1}^{k/2} x^{(l)}\wedge y^{(l)}\, , \quad \mbox{that is,} \quad
M= {\cal X}^T{\cal Y}-{\cal Y}^T{\cal X} \, , \label{r5.87} \\
{\cal X}^T=(x^{(1)}\, \cdots \, x^{(k/2)})\, , \quad {\cal Y}^T=(y^{(1)}\, \cdots \,
y^{(k/2)})\, . % \qquad {\cal X}, {\cal Y}\in {\cal V}(k/2,m)\, .
\nonumber
\end{gather}
Under the above conditions, 
the set of $k\times m$ matrices ${\cal Z}=( x^{(1)}\,y^{(1)}\,\cdots\, x^{(k/2)}\,y^{(k/2)})^T$
forms the {\it extended \/}  Stiefel variety $\bar{\cal V}(k,m)$.
In contrast to  the standard Stiefel variety, for each index $l$,
the absolute values  $|x^{(l)}|=|y^{(l)}|$ are not fixed. 
Thus, $\bar{\cal V}(k,m)$ is of dimension $km-k^2/2$, and the $k\times m$ components of $\cal Z$
play the role of excessive coordinates on it. 

Let 
$$
{\omega}=\mbox{tr}\, (d {\cal X} \wedge d {\cal Y}^T)
=\sum_{l=1}^k \sum_{i=1}^m \, d x^{(l)}_i \wedge d\,y^{(l)}_i\, 
$$
be the canonical symplectic structure on the space 
${\mathbb R}^{km}=\widetilde x^{(1)},y^{(1)},\cdots , x^{(k/2)},y^{(k/2)})$
and let $\bar\omega$ denote the restriction of 2-form $\omega$ onto 
$\bar{\cal V}(k,m)\subset{\mathbb R}^{km}$. The latter subvariety
is defined by conditions
\begin{gather} \label{constraints}
 \langle x^{(l)}, x^{(l)}\rangle -  \langle y^{(l)}, y^{(l)}\rangle =0, \quad  
\langle x^{(l)}, x^{(s)}\rangle 
= \langle y^{(l)}, y^{(s)}\rangle =0, \quad
{\cal Y}{\cal X}^T=0, \\
l,s=1,\dots, k/2,  \nonumber
\end{gather}
which consist of $k^2/2$ independent scalar equations $f_s(x,y)=0$. 
The matrix of standard Poisson brackets of the constraint functions $f_s$ in 
${\mathbb R}^{km}$ is nondegenerate. 
It follows that 2-form $\bar\omega$  is also nondegenerate
and the extended Stiefel variety is symplectic.

Since the vectors  are fixed in space, in the frame attached to the top they satisfy the Poisson
equations $\dot x^{(l)}=-\Omega x^{(l)}$, $\dot y^{(l)}=-\Omega y^{(l)}$,
$\Omega\in so(m)$, which imply
\begin{equation}
\dot{\cal X}={\cal X}\Omega\, , \quad \dot{\cal Y}={\cal Y}\Omega \, .
\label{r5.88}
\end{equation}
As above, we put $\Omega=a M+M a$, $a=\mbox{diag} (a_1,\dots,a_m)$
and define a dynamical system on $\bar{\cal V}(k,m)$, which is generated by
(\ref{r5.88}), (\ref{r5.87}):
\begin{equation}
\begin{aligned}
\dot{\cal X} &={\cal X}[ a({\cal X}^T {\cal Y}-{\cal Y}^T{\cal X})
+{\cal X}^T{\cal Y}a],  \\
\dot{\cal Y} &={\cal Y} [ a ({\cal X}^T{\cal Y}-{\cal Y}^T{\cal X})
-{\cal Y}^T{\cal X} a ] \, .  \end{aligned} \label{r588}
\end{equation}

\begin{theorem}\label{tr5.25}\begin{description}
\item{\rm 1)} Under the substitution $(\ref{r5.87})$
solutions of the system $(\ref{r588})$ give rank $k$ solutions of the Frahm--Manakov
system (\ref{FM_again}) on $so^*(m)$. 

\item{\rm 2)}
Up to the action of the discrete group generated by reflections
$({\cal X},{\cal Y})\to (-{\cal X},-{\cal Y})$, the system $(\ref{r588})$
is described by $k\times k$ Lax pair with rational parameter $\lambda$
\begin{gather}
\dot L(\lambda)=[\, L(\lambda), A(\lambda)\,]\, , \qquad L,A\in {\rm sp}(k/2), \quad
\lambda\in{\mathbb C} \, , \label{r5.89} \\
L=\begin{pmatrix}-{\cal X} (\lambda{\bf I}-a)^{-1} {\cal Y}^T &
-{\cal X} (\lambda{\bf I}-a)^{-1} {\cal X}^T \\
{\cal Y} (\lambda{\bf I}-a)^{-1} {\cal Y}^T &
{\cal Y} (\lambda{\bf I}-a)^{-1} {\cal X}^T \end{pmatrix}
\equiv \sum_{i=1}^m  \frac{ {\cal N}_i} {\lambda-a_i}\, , \label{dual1} \\
{\cal N}_i=\begin{pmatrix} {\bar x}_i {\bar y}_i^T & -{\bar x}_i {\bar x}_i^T \\
-{\bar y}_i {\bar y}_i^T & -{\bar y}_i {\bar x}_i^T \end{pmatrix}, \nonumber  \\
A=\begin{pmatrix}
{\cal X} (a+\lambda{\bf I}){\cal Y}^T  & {\cal X} (a+\lambda{\bf I}){\cal X}^T \\
-{\cal Y} (a+\lambda{\bf I}) {\cal Y}^T & -{\cal Y} (a+\lambda{\bf I}){\cal X}^T
\end{pmatrix}\,, \label{AA}
\end{gather}
where $\bar x_i=\left( x_i^{(1)},\dots,x_i^{(k/2)} \right)^T$
(respectively $\bar y_i=\left( x_i^{(1)},\dots,x_i^{(k/2)}\right)^T)$ is $i$-th column of $\cal X$
{\rm (}respectively of $\cal Y)$, and ${\bf I}$ is the unit $m\times m$ matrix.
\end{description}
\end{theorem}

\noindent{\it Proof.} The first statement follows directly from
the derivation of the system $(\ref{r588})$. Further, we calculate the derivative 
$\dot L(\lambda)$ by virtue of equations (\ref{r588}).
In view of matrix relations 
$(\lambda{\bf I}-a)^{-1} a=\lambda(\lambda{\bf I}-a)^{-1}-{\bf I}$ and 
${\cal Y}{\cal X}^T={\cal X}{\cal Y}^T=0$,  the derivative  coincides 
with the commutator in $(\ref{r5.89})$. $\boxed{}$
\medskip

\noindent{\bf Remark 2.1.} 
Notice that the entries of matrices 
$$
\Phi(\lambda)L(\lambda), \quad \Phi(\lambda)A(\lambda), \quad \mbox{where} \quad
\Phi(\lambda)=(\lambda-a_1)\cdots (\lambda-a_n)
$$  
are polynomials in $\lambda$, and, under the substitution (\ref{r5.87}),
the coefficients of the characteristic polynomial $| \Phi(\lambda) L(\lambda)-w{\bf I}|$
can be expressed in terms of $M_{ij}$ only as follows 
\begin{gather}
|w{\bf I}-L(\lambda)|=w^k +\sum_{l=2}^k w^{k-l} \Psi^{l-1}(\lambda)\, 
\widetilde {{\cal I}}_{l}(\lambda,M) \, , \qquad l=2,4,\dots,k, \nonumber \\
\widetilde {\cal I}_{l}(\lambda,M)=\sum^{m}_{I}
\frac{\Phi(\lambda) }{ (\lambda-a_{i_1})\cdots(\lambda-a_{i_l})}\,|M|^{I}_{I} \, ,
\label{cal_I}
\end{gather}
where $|M|^{I}_{I}$ are diagonal minors of order $l$ corresponding to
multi-indices $I=\{i_1,\ldots i_k\}\subset\{1,\dots,m\}$, $1\le i_1<\cdots<i_l\le m.$
Notice that the leading coefficients $H_{l,m-l}(M)=\sum^{m}_{I} \,|M|^{I}_{I}$ form
a complete set of Casimir functions on $so^*(m)$. 
\medskip

The $k\times k$ matrix $L(\lambda)$ in (\ref{dual1}) belongs to a wide class of Lax operators of the form 
$$
Y+\sum_{i=1}^n \frac{ G_i^T F_i}{\lambda-a_i},  
$$
where $Y\in gl(k)$ is a constant matrix and $G_i, F_i$ are $k_i\times k$ matrices.
Such Lax matrices can be regarded as images of moment maps to the loop algebra 
$\widetilde {gl}(k)$, and integrable systems generated by them 
have been studied in the series of papers \cite{AHP,AHH_CMP,AHH_Lett,Harnad1}
in connection with the duality to so called rank $k$ perturbations of
constant diagonal matrices of dimension $n\times n$ (following Moser \cite{Moser_Chern}). 
In particular, the  $k\times k$  Lax matrix (\ref{dual1}) 
is dual to the  $n\times n$ Lax matrix in the Manakov representation,
\begin{equation}
{\cal L} (\mu) = a +\frac 1\mu ({\cal X}^T{\cal Y}-{\cal Y}^T{\cal X} )\equiv 
  a +\frac 1\mu M,  \label{2.17} 
\end{equation}
in the sense that under the relation (\ref{r5.87})  
the  spectral curves $|L(\lambda)- \mu {\bf I}|=0$ and
$|{\cal L}(\mu)- w{\bf I}|=0$ are birationally equivalent and 
the parameter $\lambda$ plays the role of the eigenvalue parameter for (\ref{2.17}).
The characteristic polynomials of the dual Lax matrices are related by the 
Weinstein--Aronjan formula (see \cite{AHP}). 
\medskip

\noindent{\bf Remark 2.2.} The matrix $A(\lambda)$ in (\ref{AA}) can be represented in form
$$
A(\lambda)\, =[\, \lambda^{-m+2} \Phi(\lambda) L(\lambda) \, ]_+ + (a_1+\cdots +a_n) L_0 ,  \qquad
L_0=\begin{pmatrix}  0 &  {\cal X} {\cal X}^T  \\
                                    -{\cal Y} {\cal Y}^T &  0 \end{pmatrix}
$$
where $[\;\; ]_+$ denotes the polynomial part in $\lambda$ of the expression. Notice that the
Lax equation $\dot L=[L, L_0]$ describes the vector flow
\begin{equation}  \label{rot}
\dot{ x^{(l)}} =\langle x^{(l)},x^{(l)}\rangle y^{(l)}, \quad \dot{ y^{(l)}}
=- \langle y^{(l)},y^{(l)}\rangle x^{(l)}, \qquad
l=1,\dots, k/2 .
\end{equation}
For each index ${\cal R}_l$, equations  (\ref{rot})
generate rotations ${\cal R}_l$ in 2-planes spanned by the vectors $x^{(l)},y^{(l)}$, 
which leave the momentum $M$ invariant.

\medskip
Let $\overline{\{\cdot, \cdot \}}$ be the Poisson bracket on $\bar{\cal V}(k,m)$ obtained as
the Dirac restriction of the standard bracket in  ${\mathbb R}^{km}$. 
Symplectic properties of our system are descibed by 

\begin{proposition} \label{STAND}
The dynamical system $(\ref{r588})$ on $\bar{\cal V}(k,m)$  is Hamiltonian 
with respect to $\overline{\{\cdot, \cdot \}}$ with the Hamilton function 
$\bar H ({\cal X}, {\cal Y}) =-\frac 14 {\rm tr}(M^2 ({\cal X}, {\cal Y}) A)$. 
In the abundant coordinates ${\cal X}, {\cal Y}$ it admits the canonical representation
\begin{gather} \label{canonical}
\dot x_i ^{(l)}  = \frac {\partial \bar H}{\partial  y_i ^{(l)} } \bigg |_{\bar{\cal V}(k,m)}\, , \quad
\dot y_i ^{(l)}  =  -\frac {\partial \bar H}{\partial x_i^{(l)} }  \bigg |_{\bar{\cal V}(k,m)}\, ,  \\ 
i=1,\dots, n, \quad  l=1,\dots,k/2. \nonumber
\end{gather}
\end{proposition}

\noindent{\it Proof.\/} The equivalence of equations (\ref{canonical}) and  (\ref{r588}) on 
 $\bar{\cal V}(k,m)$ is verified by direct calculations.  
Next, according to the Dirac formalizm, the standard bracket and
$\overline{\{\cdot, \cdot \}}$ are different by terms containing $\{ f_s, \bar H \}$.
The latter equal zero since the constraint functions $f_s$ given by (\ref{constraints}) 
are invariants of the flow generated by $\bar H ({\cal X}, {\cal Y})$  on  ${\mathbb R}^{km}$. 
Hence, equations (\ref{r588}) or (\ref{canonical})  are Hamiltonian 
with respect to $\overline{\{\cdot, \cdot \}}$. $\boxed{}$
\medskip


Rotations ${\cal R}_l$ given by (\ref{rot}) are generated by the Hamiltonians $H_l(x,y)$, 
the restrictions of the functions 
$\frac 12 \langle x^{(l)},x^{(l)}\rangle \langle y^{(l)},y^{(l)}\rangle$
on $\bar{\cal V}(k,m)$. 
Clearly, these functions are first integrals of the system (\ref{r588}) and moreover 
they commute with  $H$. 

Let us fix the values of the Hamiltonians by putting 
$$
\langle x^{(l)},x^{(l)}\rangle =\langle y^{(l)},y^{(l)}\rangle =h_l, \quad h_l={\rm const}\ne 0, \quad l=1,\dots,k/2.
$$
These conditions define the customary Stiefel variety ${\cal V}(k,m)$. 
Under the substitution (\ref{r5.87}), the factor variety 
${\cal V}(k,m)/ \{ {\cal R}_1,\dots, {\cal R}_{k/2}\}$ coincides with 
a rank $k$ coadjoint orbit ${\cal S}_h^{(k)}\subset so^*(m)$ of dimension $k\left(m-\frac k2\right)-k$, 
which is parameterized by the constants $h_1,\dots,h_{k/2}$. Notice that
$M^2=h^2_1+\cdots +h^2_{k/2}$. 

\begin{theorem} \begin{description} 
\item{\rm 1)} Under the map $\bar{\cal V}(k,m) \to {\cal S}_h^{(k)}$, 
the Lie--Poisson bracket on  ${\cal S}_h^{(k)}\subset so^*(m)$  is the 
push-forward of the bracket $\overline{\{\cdot, \cdot \}}$.  
\item{\rm 2)}
The Poisson (Marsden--Weinstein)  reduction of the system (\ref{r588}) obtained by fixing values of $H_l(x,y)$ 
and by factorization by ${\cal R}_l$,  $l=1,\dots,k/2$ coincides with  the restriction of the
Frahm--Manakov system with Hamiltonian \\
$H(M)=\frac 12 \sum_{i\le j}(a_i+a_j) M_{ij}^2$ onto the orbit  ${\cal S}_h^{(k)}$. 
\end{description} 
\end{theorem}

\noindent{\it Proof.\/} 1). In view of (\ref{poinsot}),  (\ref{r5.87}), 
$$
\{ M_{ij}({\cal X}, {\cal Y}),  M_{kl}({\cal X}, {\cal Y}) \} 
=\{  M_{ij},  M_{kl}\}_{so(n)} ({\cal X}, {\cal Y})  ,
$$
i.e., the canonical bracket on ${\mathbb R}^{km}$ is the pull-back of  
the  bracket $\{\cdot, \cdot\}_{so(n)}$ on ${\cal S}_h^{(k)}\subset so^*(m)$.
On the other hand, on $\bar{\cal V}(k,m)$, 
$$
\{ M_{ij}({\cal X}, {\cal Y}),  M_{kl}({\cal X}, {\cal Y}) \} 
=\overline{  \{ M_{ij}({\cal X}, {\cal Y}),  M_{kl}({\cal X}, {\cal Y}) \}  } ,
$$
since for any $i,j,s$,  $\{ M_{ij}({\cal X}, {\cal Y}),  f_s ({\cal X}, {\cal Y}) \}=0$. 
This proves item  1). 

 2). By item  1) and Proposition \ref{STAND}, 
the  Poisson reduction of system (\ref{r588})  onto ${\cal S}_h^{(k)}$ is 
described by the Lie--Poisson bracket $\{\cdot, \cdot\}_{so(n)}$ and 
the Hamiltonian 
$H(M)=\bar H ({\cal X}, {\cal Y})= \sum_{i\le j}(a_i+a_j) M_{ij}^2$,
i.e., it is the corresponding restriction of the Frahm--Manakov system. $\boxed{}$
\medskip

The reduced system on the orbit ${\cal S}_h^{(k)}$  is integrable 
% as the restriction of the FM system on  $so^*(m)$  
and its generic invariant manifolds are tori of dimension
$\frac 12 {\rm dim\,} {\cal S}_h^{(k)}$  (see, e.g., \cite{MiFo}). 
On the other hand, the preimage of a generic point 
$M\in {\cal S}_h^{(k)}$ in ${\cal V}(k,m)$ is a
$k/2$-fold product of circles $S^1 \times \cdots\times S^1$
(in the complex case ${\mathbb C}^*\times \cdots \times {\mathbb C}^*$). 
This implies that the original system on $\bar{\cal V}(k,m)$ has generic invariant tori of dimension
$\frac 12 {\rm dim\,} {\cal S}_h^{(k)} + k/2=(m-k/2)k/2$,
i.e., a half of dimension of the symplectic manifold  $\bar{\cal V}(k,m)$.
Hence, the original system $(\ref{r588})$ is also integrable. 

To get a global view on the above manifolds,
we represent them in the following diagram, with the dimension 
indicated above, where arrows denote the corresponding relations (restrictions or factorizations).
$$
\begin{CD}
{\mathbb R}^{km}\quad @ > f_s=0 >>
\bar{\cal V}(k,m) \quad @ > |x ^{(l)}|^2=|y ^{(l)}|^2=h_l >> 
\quad {\cal V}(k,m)  \qquad @ > {\cal R} >> \quad {\cal S}_h^{(k)}
\end{CD}
$$
$$
\boxed{k\times m} \quad \quad \boxed{k\left(m-\frac k2\right)} \qquad \quad
\boxed{ k\left(m-\frac k2\right)-\frac k2} 
\qquad \boxed{k\left(m-\frac k2\right)-k}
$$
\medskip

\noindent{\bf Remark 2.3.} 
In the case of maximal rank $k$ ($k=m$ or $k=m-1$),
when ${\cal S}_h^{(k)}$ is a generic coadjoint orbit ${\cal S}_h$, the Stiefel variety 
${\cal V}(k,m)$ is isomorphic to the group $SO(m)$. Then the following commutative diagram 
holds
$$
\begin{CD}
\bar{\cal V}(k,m) @ > |x ^{(l)}|^2=|y ^{(l)}|^2=h_l >> SO(m) \\
       @  V   {\cal R} VV        @  V   {\cal R} VV \\
so^*(m) @ > H_{l,m-l}(M)=c_l   >> {\cal S}_h \, ,
\end{CD}
$$
where the values $\{c_l\}$  of nonzero Casimir functions $H_{l,m-l}(M)$
correspond to the constants $\{h_l\}$. The mapping
$\begin{CD} SO(m) @ > {\cal R}>> {\cal S}_h \end{CD}$ 
can be regarded as a multi-dimensional analog of the Hopf fibration
$\begin{CD} SO(3) @ >S^1 >> S^2\end{CD}$.
The restriction of the system (\ref{r588}) onto ${\cal V}(k,m)$ yields an integrable
flow on the group $ SO(m)$ which describes the motion of the Frahm--Manakov top in space
for the chosen angular momentum. 

For $m=3$ such a flow was considered in \cite{Koz_vortex, Koz_AMS} from the point of
view of its hydrodynamical interpretation. % {\bf Haar biinvariant form ???}
\medskip

\paragraph{A generalization of the Chasles theorem.} If the rank $k$ is not maximal, then the components of
${\cal X}, {\cal Y}$ themselves are not sufficient to form a complete set of coordinates
on  $SO(m)$ and to  determine the position of the top in space uniquely. 
However, in this case one can make use of the following geometric property described in \cite{Fe_AMS}.
 Let us fix a part of constants of motion by putting in (\ref{cal_I})
\begin{equation} \label{c's}
\widetilde {\cal I}_{k}(s,M)=c_{0}(s-c_{1})\cdots(s-c_{m-k}), \qquad 
c_{0}, c_{1}, \dots, c_{m-k}=\mbox{const} 
\end{equation}
and consider family of confocal cones in ${\mathbb R}^{m}=(X_1,\dots,X_n)$
\begin{equation}
\bar Q(s)= \biggl\{{X^{2}_{1}\over s-a_{1}} +\cdots+
{X^{2}_{n}\over s-a_{n}} =0\biggr\} .
\end{equation}
Let $\bar {\Lambda} \subset {\mathbb R}^{m}$ be a $k$-plane spanned by the orthogonal vectors 
$x^{(1)},y^{(1)},\dots ,  x^{(k/2)},y^{(k/2)}$. 

\begin{proposition}{\rm (\cite{Fe_AMS}).} \label{prop5.6} 
\begin{description}
\item{\rm 1).} Under the motion of the Frahm--Manakov top with constants (\ref{c's})
the  $k$-plane $\bar{\Lambda}$ is tangent to the fixed cones $\bar Q (c_{1}),\ldots,\bar Q(c_{m-k})$.
\item{\rm 2).} Let $\phi^{(\alpha)}$ be a normal vector of the cone $\bar Q(c_{\alpha})$ 
at a point of the contact line $\bar\Lambda \cap \bar Q(c_{\alpha})$. 
Then the vectors $\phi^{(1)}, \dots, \phi^{(m-k)}$ together with 
$x^{(1)}, y^{(1)}, \dots, x^{(k/2)}, y^{(k/2)}$ form an orthogonal frame in ${\mathbb R}^{m}$ 
which is fixed in space. 
\end{description}
\end{proposition}

For fixed polynomial $\widetilde {\cal I}_{k}(s,M)$, the vectors  $\phi^{(l)}$ can be calculated in 
terms of $x^{(s)}, y^{(s)}$ and, thereby, the position of the top in space is completely determined. 
Proposition \ref{prop5.6} defines a single-valued map  $\bar{\cal V}(k,m)\to SO(m)$ under which
generic invariant tori of dimension $(m-k/2)k/2$ on $\bar{\cal V}(k,m)$ become tori of the same
dimension on the group $SO(m)$.

Note that the above proposition generalizes the celebrated Chasles theorem on the propery of
the tangent line to a geodesic on a quadric.
% In the next section we shall use this approach to 
% describe explicitly the rotation matrix in the case $k=2$.

\paragraph{The rank 2 case.} 
 In the simplest case $k=2$ the angular momentum can be represented in form 
\begin{equation}\label{decom}
M=x \wedge y, \quad x=x^{(1)}=(x_1,\dots,x_m)^T  \, ,\quad 
y=y^{(1)}=(y_1,\dots,y_m)^T 
\end{equation}
and equations (\ref{r588})  describe a Hamiltonian system on the extended Stiefel variety  
$\bar{\cal V}(2,m)=\left\{ (x,y) \bigg|\,|x|=|y|, \; \langle x,y\rangle =0\right\}$, 
\begin{equation}
\label{bg}
\begin{aligned}
\dot x &=-\langle y, a x\rangle x +\langle x, a x\rangle y + a y \langle x,x\rangle , \\
\dot y &=-\langle y, a y\rangle x +\langle x,a y\rangle y -a x \langle y,y\rangle  
\end{aligned}
\end{equation}
with the Hamiltonian 
$$ \bar H= \frac 12  \langle x,a x\rangle \langle y,y\rangle 
-\langle a x,y\rangle \langle x,y\rangle  
+ \frac 12  \langle y,a y\rangle   \langle x,x\rangle 
= \frac 12  \sum_{i<j}^m (a_i+a_j) M_{ij}^2 .
$$ 

Equivalently, this system describes the evolution of fixed orthogonal vectors $x,y$ in a frame attached 
to the $m$-dimensional body.
The system admits the following $2\times 2$ Lax pair arising from (\ref{r5.89}),
\begin{gather}
\dot L(\lambda)=[\, L(\lambda), {\cal A} (\lambda)\,]\, , \label{r5.99} \\
L(\lambda)=\Phi(\lambda) \begin{pmatrix}
-\sum_{i=1}^m \frac{x_i y_i}{\lambda-a_i} &
-\sum_{i=1}^m \frac{x_i^2}{\lambda-a_i} \\ \sum_{i=1}^m \frac{y_i^2}{\lambda-a_i} &
\sum_{i=1}^m \frac{x_i y_i}{\lambda-a_i} \end{pmatrix}\, , \nonumber \\
\medskip \nonumber
{\cal A} (\lambda)=\begin{pmatrix}
-\sum_{i=1}^m (\lambda+ a_i){x_i y_i}  & -\sum_{i=1}^m (\lambda + a_i)x_i^2  \\
\sum_{i=1}^m (\lambda + a_i)y_i^2 &
\sum_{i=1}^m  (\lambda + a_i) {x_i y_i} \end{pmatrix}\, ,  \nonumber \\
\Phi(\lambda)=(\lambda-a_{1})\cdots(\lambda-a_{m}),  \nonumber 
\end{gather}

The Lax representation (\ref{r5.99}) was first indicated in
\cite{AHH_Lett}, where it was shown to be dual to an $n\times n$ Lax pair 
for the rank 2 case found by Moser in \cite{Moser_Chern}.  


In view of relation (\ref{decom}), the characteristic polynomial  
$|L(\lambda)-\mu {\bf I} |$ for (\ref{r5.99}) can be written in form
$\Phi(\lambda) \widetilde {\cal I}_2(\lambda,M) +\mu^2$,
where $\widetilde {\cal I}_2(\lambda,M)$ is the family of quadratic integrals defined in (\ref{cal_I}),
\begin{align}
\widetilde {\cal I}_2(\lambda,M) &
=\sum_{i<j} \frac { \Phi(\lambda)}{(\lambda-a_{i})(\lambda-a_{j})}M_{ij}^2
\nonumber \\
& =\lambda^{m-2} H_{2,m-2}(M)+\lambda^{m-3} H_{2,m-3}(M)+\cdots+H_{20}(M) .
\label{m2.13} 
\end{align}
Notice that $H_{2,m-2}(M)= \sum_{i<j}^m M_{ij}^2=(y,y)(x,x)$ is a Casimir function
of the standard Lie--Poisson bracket on $so^*(m)$. With respect to the Poisson bracket 
$\overline{\{\, , \, \} }$ on $\bar{\cal V}(2,m)$, this function  generates permanent
rotations of the top in the fixed 2-plane $\bar\Lambda ={\rm span}(y,x)$,
which leave the components of $M$ invariant. 

Let us fix the constants of motion by putting
\begin{equation}
\label{I_2}
\widetilde {\cal I}_2(\lambda,M) =c_0(\lambda-c_1)\cdots (\lambda -c_{m-2}) , \qquad 
c_0, c_1,\dots, c_{m-2}=\mbox{const}. 
\end{equation}
This defines hyperelliptic spectral curve in ${\mathbb C}^2=(\lambda,\mu)$ of genus $g=m-2$
\begin{equation} \label{hypp}
{\cal C}=\{ \mu^2= -c_0\,  \Phi(\lambda)\, (\lambda-c_1)\cdots (\lambda -c_{m-2})\} \, .
\end{equation}

As noticed in \cite{Acta_bill}, the real generic $(g+1)$-dimensional invariant 
tori of the system can be extended to open subsets of generalized Jacobian varieties 
Jac$({\cal C},\infty_{\pm})$, which are extensions of the customary $g$-dimensional 
Jacobian Jac$({\cal C})$ by ${\mathbb C}^*$ and which can be regarded as the factor of
${\mathbb C}^2$ by the lattice generated by $(2g+1)$ independent period vectors of 
$g$ holomorphic differentials $\bar \omega_1\dots,\bar\omega_g$ and a meromorphic differential
of the third kind  $\varOmega_{\infty_{\pm}}$ having a pair of simple poles at the infinite
points $\infty_{\pm}$ on the curve $\cal C$. 

The coefficients of the matrix polynomial $L(\lambda)$ are meromorphic functions on
Jac$({\cal C},\infty_{\pm})$, whereas the components of the momentum $M_{ij}$ and 
the normal vectors $\phi^{(\alpha)}$ are meromorphic on a covering of the
Jacobian Jac$({\cal C})$ itself (the ${\mathbb C}^*$-extension is factored out by
the action of ${\cal R}=SO(2)$).

In the classical case $m=3$ the curves $\cal C$ become elliptic ones and 
generic invariant tori in $\bar{\cal V}(2,3)$ and in $SO(3)$ are 2-dimensional. 
An explicit solution for the components of the rotation matrix in terms of theta-functions
and exponents was first given in \cite{Jac_corps} (see also \cite{Whitt}).
Since now rank $M=$2 in the generic case, the above commutative diagram takes the form
$$
\begin{CD}
\bar{\cal V}(2,3) @ > |x|^2=|y|^2=h >> SO(3) \\
       @  V   SO(2) VV        @  V   SO(2) VV \\
so^*(3) @ > \langle M,M\rangle =h^2   >> S^2_h 
\end{CD}
$$
$S^2_h$ being the coadjoint orbit (2-dimensional sphere) corresponding to the constant $h$.

\section{B\"acklund transformation on $\bar{\cal V}(2,3)$, $SO(3)$ \\ 
and discretization of the clasical Euler top}

A first integrable discretization of the $m$-dimensional Euler--Manakov top was 
constructed in \cite{Ves, MosVes} by the method of factorization of matrix polynomials. 
It was represented by the correspondence 
$(\Omega,M)\to (\widetilde  \Omega, \widetilde  M)$, $\Omega\in SO(m)\,, M\in so^*(m)$, which, in
our notation reads
\begin{equation} \label{step}
M=\Omega^T A- A\Omega, \quad  \widetilde  M=\Omega M\Omega^T .
\end{equation}
Given $\widetilde  M$,  the new matrix $\widetilde \Omega$ is found from equation 
$\widetilde  M=\widetilde \Omega^T A- A\widetilde \Omega$, whose solution is not unique. 

 In given section we describe a symplectic map 
$\bar{\cal B}_{\lambda*}  \; : \bar{\cal V}(2,3) \to \bar{\cal V}(2,3)$, 
$\bar{\cal B}_{\lambda*}(x,y) =(\widetilde  x,\widetilde  y)$
governed by an arbitrary parameter $\lambda^*\in {\mathbb C}$,  which preserves the first 
integrals of the continuous system (\ref{bg}) and whose restriction 
to each generic complex torus, generalized Jacobian Jac$({\cal C},\infty_{\pm})$, is given by shift by the
2-dimensional vector 
$$
S=\int_{E_-}^{E_+} 
\left(\frac{d\, \lambda}{\mu} ,\frac{\lambda\, d\, \lambda}{\mu} \right)^T, \qquad 
E_{\pm}=(\lambda^*,\pm \mu^*), 
$$
$E_{\pm}$ being involutive points on the elliptic spectral curve, 
the simplest case of (\ref{hypp}),
$$
{\cal C}= \{\mu^2= -c_0\, (\lambda-a_1)(\lambda-a_2)(\lambda-a_3)\, (\lambda-c_1) \}.
$$
Note that $S$ is a correctly defined vector in the generalized Jacobian:
under a change of integration path on $\cal C$ it increases by a period vector
of Jac$({\cal C},\infty_{\pm})$. 
We also emphasize that here $\lambda^*$ is a constant parameter, whereas 
the conjugated coordinate $\mu^*$ depends on the equation of the curve. 

\paragraph{B\"acklund transformation on $\bar{\cal V}(2,3)$.} 
As shown in \cite{Sabaudia} by applying an addition theorem for a class of meromorphic functions on
generalized hyperelliptic Jacobians, such a map admits intertwining relation (discrete Lax pair)
\begin{gather}
\widetilde  L(\lambda) M(\lambda|\lambda^*)=M(\lambda|\lambda^*) L(\lambda), \label{bak4.8} \\ 
M(\lambda|\lambda^*)=
\begin{pmatrix} -\alpha (\lambda-\lambda^*)+ \beta & 1 \\
-\beta^2 &   -\alpha (\lambda-\lambda^*)-\beta  \end{pmatrix}, \nonumber \\
\begin{aligned}
\beta(\lambda^*) & = - \frac{\mu^*+L_{11}(\lambda^*)}{L_{12}(\lambda^*)}\equiv 
- \frac{\mu^*+ a_1^* a_2^* a_3^* \langle x,(a-\lambda^* {\bf I} )^{-1}y\rangle }
{a_1^* a_2^* a_3^* \langle x,(a-\lambda^* {\bf I} )^{-1}x\rangle }, \\
\alpha & =\frac{\partial\beta(\lambda)}{\partial\lambda} \bigg |_{\lambda=\lambda^*} ,
\quad a^*_i = a_i -\lambda^*, 
\end{aligned}
\label{ab}
\end{gather}
where $L(\lambda)$ is defined in  (\ref{r5.99}) and $\widetilde  L(\lambda)$ depends on the new variables 
$\widetilde  x, \widetilde  y$ in the same way as $L(\lambda)$ depends on $x, y$.  
In view of (\ref{m2.13}) for $m=3$, 
\begin{equation} \label{*}
\mu^*=\sqrt{\Phi (\lambda^*) \sum_{k=1}^3 a_k^* (x_i y_j- x_j y_i)^2}\, , \qquad 
(i,j,k)=(1,2,3).
\end{equation}

Now putting in (\ref{bak4.8}) subsequently  $\lambda=a_1,a_3,a_3$
and calculating the matrices $M(a_i) L(a_i|x, y) M^{-1}(a_i)$, we find
\begin{align}
{\widetilde  x}_i^2 &=\frac{ (y_i +\beta x_i+ \alpha (a_i-\lambda^*) x_i)^2} { \alpha^2 (a_i-\lambda^*)^2}, 
\nonumber \\
{\widetilde  y}_i^2 &=\frac{ (\beta y_i +\beta^2 x_i - \alpha (a_i-\lambda^*) y_i)^2 }
{ \alpha^2 (a_i-\lambda^*)^2},  \label{sqared} \\
\widetilde  x_i \widetilde  y_i &=- \frac{(y_i +\beta x_i+ \alpha (a_i-\lambda^*) x_i) \,
(\beta y_i +\beta^2 x_i- \alpha (a_i-\lambda^*) y_i)} {\alpha^2 (a_i-\lambda^*)^2}. \nonumber
\end{align}
From here the new variables can be recovered up to the action of the group generated by reflections
$(\widetilde  x_i, \widetilde  y_i)\to (-\widetilde  x_i, -\widetilde  y_i)$. Imposing the condition of the existence of 
a continuous limit (see below), we choose the following relations
\begin{gather}
\widetilde  x_i- x_i= \frac{ y_i +\beta x_i}{\alpha (a_i-\lambda^*)}, \quad
\widetilde  y_i- y_i= -\frac{\beta(y_i+\beta x_i)}{\alpha (a_i-\lambda^*)}, \qquad 
i=1,2,3 .  \label{4.10}  
\end{gather}
These expressions together with (\ref{ab}), (\ref{*}) describe the map
 $\bar{\cal B}_{\lambda*}  \; : \bar{\cal V}(2,3) \to \bar{\cal V}(2,3)$ 
in an {\it explicit\/} form. 
Since a generic parameter $\lambda^*$ corresponds to two values of $\mu^*$,
the map is generally two-valued. 
\medskip

\paragraph{Geometric model.} 
The restriction of the map onto the group $SO(3)$ admits a transparent geometric interpretation,
which can be regarded as a ``discrete version'' of the kinematic Poinsot model (see, e.g., \cite{Whitt}).
Namely, let $|x|=|y|=1$ and let 
$$R=|| x \; y \;\, x\wedge  y || \in SO(3)$$ 
be rotation matrix defining a position of a rigid body 
in space. We attach to the body a cone 
$K_2=\{(X,(a-\lambda^* {\bf I} )^{-1}X)=0\}$, which is fixed in the body
frame $(X_1,X_2,X_3)$, and assume that 
\begin{equation}
\label{real}
0< a_1 < a_2 <a_3,  \quad a_1 < \lambda^* < a_2 \quad \mbox{or} \quad a_2 < \lambda^* < a_3 .
\end{equation}
Under these conditions the cone is real and regular. 
Let $\Pi$ be 2-plane spanned by $x,y$, which is thus fixed in space and orthogonal 
to the momentum vector $M=x\wedge  y$. Assume also that $x,y$ are such that  $\Pi$ has 
a nonempty real intersection with the cone $K_2$ along lines $L_1, L_2$. 
One can show that under this condition the coordinates $\mu^*$ defined in (\ref{*})
and the parameters $\alpha, \beta$ are real. 

\begin{theorem} \label{proj_bill}
Let $K_1=\{(X,(a- h {\bf I})^{-1}X)=0\}$, $h=\textup{const}$ be a unique cone attached to the body
such that it is confocal to $K_2$  and tangent to the fixed plane $\Pi$. 
Then then new position of the body defined by the rotation matrix 
$\widetilde  R=|| \widetilde  x\;\widetilde  y\;\,\widetilde  x\wedge \widetilde y ||$ and expressions (\ref{4.10}) 
is obtained from the original position by rotating the cones
$K_1, K_2$ about axis $L_1$ or $L_2$ until $K_1$ again touches $\Pi$. 
\end{theorem}

This geometric construction is illustrated on Figure 1. 
In the new position $\widetilde  R$ determined by rotation about $L_2$, the cone $K_2$ intersects $\Pi$
along $L_2$ and another line $L_3$. Then the next iteration is generated by rotation about $L_2$ or $L_3$.
The two-valuedness of the map 
$\bar{\cal B}_{\lambda*}$  
is now related to the possibility of rotation about two different axes in ${\mathbb R}^3$. 
It follows that $N$-th iteration of the map is only $(N+1)$-valued, not $2^N$-valued. 
By fixing a sign of $\mu^*$ in (\ref{*}),  $\bar{\cal B}_{\lambda*}$  becomes single-valued
and generates a sequence of  points on $SO(3)$. 

The geometric model was first proposed in \cite{Sabaudia} as a certain limit of 
a kinematical model of motion of 4-dimensional Frahm--Manakov top in space. 
\medskip

\epsfysize=7cm
\epsfxsize=13cm
\epsfbox{cc3.eps}
\centerline {Figure 1}
\medskip


\noindent{\bf Remark 3.1.} As follows from (\ref{4.10}), the map $\bar{\cal B}_{\lambda*}$
admits particular solutions, for which the vector $M=x\wedge  y$ remains to be
an eigenvector of the inertia tensor $A=\mbox{diag}(a_1,a_2,a_3)$, whereas
$x, y$ themselves rotate by a fixed angle in the plane $\Lambda$. 
Such solutions can be regarded as 
analogs of stationary permanent rotations of the classical Euler top
about its principal intertia axes.

\paragraph{Continuous limit.}
Note that when $x,y$ are chosen such that $\Pi$ is (almost) tangent to the cone $K_2$ 
($c\to \lambda^*$),
$K_1$ and $K_2$ confluent and, according to the above model,  the cone $K_1=K_2$ 
is rolling without sliding 
over the fixed plane thus giving a continuous limit motion on the group $SO(3)$. 
% of the body in space.

From the algebraic geometrical point of view, in the above limit the points $E_-, E_+$
on the spectral curve $\cal C$ come together to a branch point $E_0=(c,0)$ 
and the shift vector $S$ on the generalized Jacobian tends to zero.
Let $\epsilon$ be a small complex parameter. 
Setting $\mu^*=\epsilon$, $\lambda-\lambda^*=\mbox{const}\cdot \epsilon^2$ 
in (\ref{ab}), we have the expansions
$$
\beta=-\frac {\langle x,a^{-1}y\rangle }{\langle x,a^{-1}x\rangle}+O(\epsilon), \quad 
\alpha= \frac 1\epsilon {\varkappa}{\langle x,a^{-1}x\rangle } +O(1), \qquad 
\varkappa=\frac{\partial \mu^2(\lambda)}{\partial \lambda}\bigg |_{\lambda=\lambda^*},
$$
where $\varkappa$ is a real nonzero constant. Now we set 
$$
\widetilde  x=x+\dot x\epsilon+ O(\epsilon^2), \quad 
\widetilde  y=y+\dot y\epsilon+ O(\epsilon^2) .
$$
Substituting the above expansions into (\ref{4.10}), then matching the coefficients at $\epsilon$ 
in both sides and taking into account relation 
$$ 
\langle x, (a^*)^{-1}y\rangle ^2= \langle x, (a^*)^{-1}x\rangle 
 \langle y, (a^*)^{-1}y\rangle + O(\epsilon^2),
$$
we obtain the following differential equations describing  the limit flow on a subset
of $\bar{\cal V}(2,3)$
\begin{gather} \label{cont}
\begin{aligned}
\dot x & = \frac {\det a^*}\varkappa \bigg[\langle x, (a^*)^{-1}y\rangle  (a^*)^{-1}x 
- \langle x, (a^*)^{-1}x\rangle  (a^*)^{-1}y \bigg]    
\equiv \frac 1\varkappa  x\wedge  a^*(x\wedge  y), \\
\dot y &= \frac {\det a^*}\varkappa \bigg[\langle y, (a^*)^{-1}y\rangle (a^*)^{-1}x 
- \langle x, (a^*)^{-1}y\rangle (a^*)^{-1}x \bigg]    
\equiv \frac 1\varkappa y\wedge  a^*(x\wedge  y) , 
\end{aligned} \\
a^*=a-\lambda^* {\bf I} . \nonumber 
\end{gather}
These equations are Hamiltonian with the Hamilton function 
$$
H=\frac {\det a^*}{2\varkappa} \bigg[ \langle x, (a^*)^{-1}x\rangle 
\langle y, (a^*)^{-1}y\rangle -\langle x,(a^*)^{-1}y\rangle ^2\bigg]
\equiv  \frac 1{2\varkappa} \sum_{i=1}^3 (a_i-\lambda^*) M_i^2.
$$
Notice that this function equals zero on the limit continuous flow.
The restriction of this flow on Jac$({\cal C},\infty_{\pm})$ is tangent to 
the curve ${\cal C} \subset \mbox{Jac} ({\cal C},\infty_{\pm})$ at the point $E_0$.

The above asymptotic of $\alpha, \beta$ explains the specific choice of sign of $\widetilde  x, \widetilde  y$ made in the
passage from relations (\ref{sqared}) to the map (\ref{4.10}).
\medskip

\noindent{\it Proof of Theorem} \ref{proj_bill}. 
The condition for $\Pi$ to be tangent to the cone \\ $K_1=\{(X,(a- h {\bf I})^{-1}X)=0\}$ has the form 
$$
\sum_{k=1}^3 (h-a_k) (x_i y_j- x_j y_i)^2 \equiv {\cal I}_2(h, M)=0 .
$$
Comparing this with the family of integral (\ref{I_2}) for $m=3$, we conclude that
$h=c_1$, which is constant under the map, hence the plane spanned by $\widetilde  x, \widetilde  y$ is
again tangent to $K_1$. 

Next, any translation in $SO(3)$ is represented as a finite rotation about an axis in ${\mathbb R}^3$. 
As follows from relations (\ref{4.10}),  $\widetilde  y +\beta\widetilde  x = y +\beta x$, hence 
the line along the vector  $\ell=y +\beta x$ is invariant of the action of $\bar{\cal B}_{\lambda*}$  
on ${\mathbb R}^3$ and therefore represents the axis of such a rotation. 
Finally,  in view of (\ref{ab}), we have
\begin{align*}
\langle \ell,(a^*)^{-1} \ell\rangle &= \langle y, (a^*)^{-1} y\rangle  
-2 \frac{\mu^*+L_{11}(\lambda^*)}{L_{12}(\lambda^*)}  \langle x,(a^*)^{-1}y\rangle  \\
& \quad  + \frac{(\mu^*)^2 +2\mu^* L_{11}(\lambda^*)+L_{11}^2(\lambda^*)}
{L_{12}^2(\lambda^*)} \langle x,(a^*)^{-1}x\rangle  \\
& = \frac 1 {L_{12}(\lambda^*) } 
\left[ \langle y, (a^*)^{-1} y\rangle  \langle x,(a^*)^{-1}x\rangle 
 - \langle x, (a^*)^{-1} y\rangle ^2+ (\mu^*)^2\right] ,
\end{align*}
 which equals zero by virtue of (\ref{*}).
Hence $\langle \ell,(a^*)^{-1} \ell\rangle =0$, which imply that 
the vector  $\ell$ lies on the cone $K_2$. This establishes the theorem. $\boxed{}$
\medskip

\noindent{\bf Remark 3.2.} When the attached cone $K_{2}$ does not have real intersection with 
$\Pi=\mbox{span}(x,y)$, the coordinate $\mu^*$ is imaginary and, according to (\ref{4.10}), 
(\ref{ab}), the new values $\widetilde  x, \widetilde  y$ are complex. As a result, under the reality conditions
(\ref{real}) the map $\bar{\cal B}_{\lambda*}$  is real only on the subset 
$\bar {\mathfrak R}\subset \bar{\cal V}(2,3)$
defined by unequality 
$$
\sum_{k=1}^3 (\lambda^*-a_k) (x_i y_j- x_j y_i)^2 \le 0
$$
On the boundary of $\mathfrak R$, the map  tends to the identical one. 

\paragraph{Reduction to the coalgebra $so^*(3)$.} Under the factorization by rotations of
${\cal R}=SO(2)$, the transformation $\bar{\cal B}_{\lambda*}$  induces a map 
${\cal B}_{\lambda*}\, :\, so(3)^* \to so(3)^*$  such that 
$$
\widetilde  M \equiv {\cal B}_{\lambda*} M(x,y) =\widetilde x \wedge \widetilde y. 
$$
The latter map is correctly defined, i.e., it does not depend on a concrete choice of vectors
 $x,y$ giving the same $M$. It preserves the first integrals of the 
classical Euler top on $so^*(3)$ and its generic invariant manifolds are open subsets 
of 4-fold unramified coverings of the complex torus  Jac$({\cal C})={\cal C}$. 
The restriction of ${\cal B}_{\lambda*}$ onto 
Jac$({\cal C})$ is given by shift by the holomorphic integral
$e=\int_{E_-}^{E_+} d\lambda/\mu$, which thus depends only on the constants $c_0, c_1$.
According to a theorem in \cite{Bob_Lorb},  this implies that  the map ${\cal B}_{\lambda*}$ 
preserves the standard Lie--Poisson structure on $so^*(3)$. 

\begin{proposition} \label{reflection} 
Vectors $M, \widetilde  M$ satisfy the following symmetric relations
\begin{gather} \label{b-l-s}
\widetilde  M -M = \varkappa \, (\widetilde  M + M) \wedge  a (\widetilde  M + M) , \\
\varkappa = \sqrt{2 \langle  aM+ a\widetilde  M, aM+ a\widetilde  M\rangle  }\, 
\frac {\sqrt {1- \langle M,\widetilde M\rangle/c_0}}
{\sqrt {1+\langle M,\widetilde M\rangle /c_0} }\, ,
\label{kappa} \\
 \langle M, a^* M\rangle =\langle \widetilde  M, a^* \widetilde  M\rangle 
= - \langle M,a^*\widetilde M\rangle , \label{bill}
\end{gather}
where, as above, $c_0=\langle M,M\rangle =\langle \widetilde  M,\widetilde M\rangle$.
\end{proposition}

Relation (\ref{b-l-s}) was previously obtained by  another method  in \cite{Bob_Lorb}, 
as an implicit map describing a Poisson discretization of the Euler top in $so^*(3)$. 
\medskip

\noindent{\it Proof of Proposition} \ref{reflection}. 
In view of relations  (\ref{4.10}),  we find
\begin{equation} \label{times}
\widetilde x \wedge \widetilde y =\widetilde M
= x\wedge  y+ \frac 1\alpha \left(a^*\right)^{-1} \ell \wedge \ell, 
\end{equation}
where, as  above,  $\ell=y+\beta x$,  $a^*=a-\lambda^*{\bf I}.$
Note that vector $\left(a^*\right)^{-1} \ell$ is normal to the cone $K_2$ at a point of the intersection
line $L_2$ or  $L_1$.  Hence, $\widetilde  M -M$ is orthogonal to $\left(a^*\right)^{-1}\ell$ and $\ell$.
Next, since $\ell$ lies in the planes $\Pi, \widetilde  \Pi$, this vector is orthogonal to $M, \widetilde  M$.
This, together with the equality $|M|=|\widetilde  M|$ implies that the sum $\widetilde  M +M$ is
parallel to $\left(a^*\right)^{-1} \ell$ and  $a^*(\widetilde  M +M)$ is parallel to $\ell$. 
As a result, (\ref{times}) implies (\ref{b-l-s}).

To find factor $\varkappa$, we first introduce angle $\phi$ between vectors 
$\widetilde  M$ and $M$. Since $|M|=|\widetilde  M|$, the vectors $\widetilde  M -M$ and $\widetilde  M+M$ 
are orthogonal, and we have
$$
|\widetilde  M -M|= \frac 12 |\widetilde  M+M|\tan \frac\phi 2 
\equiv \frac 12|\widetilde  M+M| \frac {\sqrt {1-\langle M,\widetilde M\rangle/c_0 }}    
{\sqrt {1+\langle M,\widetilde M\rangle/c_0}} \, .   
$$
On the other hand, since $a^*(\widetilde  M+M)$ is orthogonal to $\widetilde  M, M$, from 
(\ref{b-l-s}) and the properties of the vector product we deduce 
$$
|\widetilde  M -M|= \varkappa \,|\widetilde  M+M|\, |a^*(\widetilde  M+M)|. 
$$
Comparing the right hand sides of the above two relations, we obtain (\ref{kappa}).

The first equality in (\ref{bill}) holds because the map  ${\cal B}_{\lambda*}$ preserves the first integrals
of the Euler top. Next, since the vector $a^*(\widetilde  M +M)$ lies on the cone
$K_2$, we have $((\widetilde  M +M), a^*(\widetilde  M +M))=0$. Expanding this and using the
 first equality in (\ref{bill})  yields the second equality. $\boxed{}$
\medskip

\noindent{\bf Remark 3.3.} 
 The fact that the difference $\widetilde  M -M$ is orthogonal to $\left(a^*\right)^{-1}\ell$ and $\ell$ implies 
that {\it the angle between the normal vector $\left(a^*\right)^{-1}\ell$
and the plane $\Pi$ equals the angle between $\left(a^*\right)^{-1}\ell$ and \/} $\widetilde \Pi$. 
This property can be regarded as a projective version of the Birkhoff condition of elastic impacts,
hence the geometric construction of Theorem \ref{proj_bill} illustrated in Figure 1 describes  a projective analog of
the plane  elliptic billiard. (Note that no any plane section of the cones and of the sequence of $\Pi$ 
gives such a plane billiard.)
 \medskip

To obtain the map ${\cal B}_{\lambda*}\, :\, so^*(3) \to so^*(3)$  in an explicit form, 
we use the fact that the vector $\ell$ satisfies the system of homogeneous equations
$$
\langle \ell,(a^*)^{-1} \ell\rangle=0,  \quad \langle \ell, M\rangle =0 . 
$$
One of its solutions, 
$ \bar\ell =(\bar\ell_1, \bar\ell_2, \bar\ell_3)^T$, normalized by the condition $\bar \ell_3=1$, 
has the form
\begin{gather}
\bar \ell_1= - a_1^* \frac {M_1 M_3 - \sqrt{D}\,  a_2^* M_2 }{a_1^* M_1^2 + a_2^* M_2^2},  \quad
\bar\ell_2= - a_2^*  \frac {M_2 M_3 + \sqrt{D} \, a_1^* M_1}{a_1^* M_1^2 + a_2^* M_2^2}, \quad
\bar \ell_3=1 \\
D= \frac{a_1^* M_1^2+ a_2^* M_2^2 +a_3^* M_3^2} {a_1^* a_2^* a_3^*} . \label{D}
\end{gather}
Substituting this instead of $\ell$ into (\ref{times}) and symmeterising obtained 
expressions, we arrive at the following relations
\begin{equation} \label{diff}
\begin{aligned}
\widetilde  M_1 -M_1 &=\chi (a_2 - a_3) 
\Big( \Delta_1\,  M_2 M_3 
  - \sqrt{D}\, (a^*_{1}) ^{2} M_1^3 ( a_2^* M_2^2 -a_3^* M_3^{2}) \Big) , \\
\widetilde  M_2 -M_2 &=\chi (a_3 - a_1) \Big( \Delta_2\, M_1 M_3 
- \sqrt{D}\, (a_{2}^*)^{2} M_2^{3} ( a_3^* M_3^2 -a_1^* M_1^2 ) \Big) , \\
\widetilde  M_3 -M_3 &=\chi (a_1 - a_2) 
\Big(\Delta_3 M_1 M_2 - \sqrt{D}\, (a_{3}^*)^2 M_3^{3}(a_1^* M_1^2 -a_2^* M_2^2) \Big) ,
\end{aligned}
\end{equation}
where 
$$
\Delta_i=a_{j}^* a_{k}^* M_j^2 M_k^2 - (a^*_i)^2 M_i^4 , \qquad 
(i,j,k)=(1,2,3)
$$
and $\chi$ is a common factor.

Next, multiplying the both sides of (\ref{diff}) by $a_1^*M_1, a_2^*M_2, a_3^*M_3$ 
respectively, then summing and using the second equality in (\ref{bill}),  
we find the factor $\chi$ in form
\begin{equation} \label{chi}
\chi= - \frac{2\langle M, a^*M\rangle} {\sum a_1 (a_2 - a_3) \bigg[ 
\Delta_1 M_1 M_2 M_3 
-\sqrt{D}(a^*_{1})^{2} M_1^4 ( a_2^* M_2^2 -a_3^* M_3^{2})\bigg]}\, ,
\end{equation}
where the summation in the denominator ranges over the three terms obtained by 
the cyclic permutations of indices (1,2,3). Thus the right hand sides of (\ref{diff})
are homogeneous expressions of degree -1 in $M_i$.
\medskip

We summarize our results on the map ${\cal B}_{\lambda*}$ in the following theorem.  

\begin{theorem} The map ${\cal B}_{\lambda*}\, :\, so^*(3) \to so^*(3)$  given by
(\ref{diff}), (\ref{chi}), (\ref{D}) preserves the first integrals of the continuous Euler top,
as well as the standard Lie--Poisson structure on $so^*(3)$.
Under reality conditions (\ref{real}), the map is real inside the conical domain
$$
{\mathfrak R} =\left\{ \sum_{k=1}^3 (\lambda^*-a_k) M_k^2 \le 0\right \}.
$$
Its restriction onto each regular elliptic curve ${\cal C}$ is represented by the shift by 
the holomorphic integral $e=\int_{E_-}^{E_+} d\lambda/\mu$. Near the boundary of 
${\mathfrak R}$ the energy constant $c_1$ tends to $\lambda^*$ and 
the shift $e$ tends to zero. 
The continuous limit of ${\cal B}_{\lambda*}$ coincides with the Euler equations
$\dot M=[M,aM]$.
\end{theorem}

\paragraph{Remark 3.4.}
Since generic solutions $M_1(t), M_2(t), M_3(t)$ of the classical Euler top are 
proportional to the elliptic functions
$\textup{sn}(u), \textup{cn} (u), \textup {dn}(u)$, with $u=\mbox{const}\cdot t$ 
and the map ${\cal B}_{\lambda*}$ results in adding $e$ to the argument $u$, 
the relations (\ref{diff}), (\ref{D}), (\ref{chi}) can be regarded as
a set of explicit addition formulae for these functions.
However, in this case neither the moduli of the curve $\cal C$ nor the shift parameter
are fixed: they depend on initial position of $M$ in $so^*(3)$.
 
%\paragraph{Discretization of the Euler top on the group $SO(3)$}

\begin{thebibliography}{35}

\bibitem{AHP} Adams M.R., Harnad J., and Previato E.
  Isospectral Hamiltonian flows in finite and infinite dimensions. I. 
Generalized Moser and moment maps in loop algebras.
{\it Comm. Math. Phys.\/} {\bf 117} (1988), 451--500

 \bibitem{AHH_CMP} Adams M.R., Harnad J., and Hurtubise J.
   Darboux Coordinates and Liouville--Arnold Integration in
   Loop Algebras. {\it Comm. Math. Phys.\/} {\bf 155} (1993), 385--413

 \bibitem{AHH_Lett} Adams M.R., Harnad J., and Hurtubise J.
   Darboux Coordinates on Coadjoint Orbits of Lie Algebras.
   {\it Let. Math. Phys.\/} {\bf 40} (1997), 41--57

\bibitem{Bl_Cr} Bloch, A.M. and Crouch P. Optimal control and geodesic flows. 
{\it Systems Control Lett.\/} {\bf 28} , no. 2, (1996),  65-72.


\bibitem{BCMR} Bloch, A. M., P. Crouch, J. E. Marsden and T. S. Ratiu. 
The symmetric representation of the rigid body equations and their discretization. {\it Nonlinearity} {\bf 15} (2002), no. 4, 1309--1341

\bibitem{Bob_Lorb}  Bobenko A. I., Lorbeer B., and Suris Yu.
Integrable discretizations of the Euler top
{\it J. Math. Phys.\/} {\bf 39}, no. 12 (1998), 6668--6683

\bibitem{Sovr_Geom} Dubrovin B.A., Novikov S.P., Fomenko A.T. 
Modern Geometry. Springer--Verlag, 1989 

\bibitem{Fe_AMS}  Fedorov Yu. Integrable systems, Lax representations, and confocal
   quadrics. In: {\it Dynamical systems in classical mechanics\/}, 173--199,
   Amer. Math. Soc. Transl. Ser. 2, {\bf 168}, Amer. Math. Soc., Providence, RI,
   1995

\bibitem{Acta_bill} Fedorov Yu.
   Classical integrable systems and billiards related   to generalized Jacobians.  {\it Acta Appl. Math.\/}
   {\bf 55} 3 (1999) 251--301

\bibitem{Sabaudia} Fedorov Yu. Discrete versions of some algebraic integrable systems
 related to generalized Jacobians. {\it SIDE III---symmetries and integrability
of difference equations\/} (Sabaudia, 1998), 147--160,
CRM Proc. Lecture Notes {\bf 25}, Amer. Math. Soc., Providence, RI, 2000

\bibitem{FFrahm}  Frahm F. \"Uber gewisse Differentialgleichungen.
{\it Math. Ann.\/} {\bf 8} (1874), 35--44

\bibitem{Harnad1} Harnad J. Dual Isomonodromic Deformations and 
Moment Maps in Loop Algebras. {\it Comm. Math. Phys.\/} {\bf 166} (1994), 337--365

\bibitem{Jac_corps} Jacobi K.G.
Sur la rotation d'un corps. In: {\it Gesamelte Werke\/} {\bf 2} (1884), 139--172

\bibitem{Jap} Hirota, Ryogo; Kimura, Kinji. Discretization of the Euler top. 
{\it J. Phys. Soc. Japan\/} {\bf  69} (2000), no. 3, 627--630

\bibitem{Koz_vortex}  Kozlov V.V.  The vortex theory of the top.
{\it Vestn. Moskov. Univ., Ser. I. Mat. Mekh.\/} No. 4 (1990) 56--62  (Russian)

\bibitem{Koz_AMS} Kozlov V. V. Hydrodynamics of noncommutative integration of
Hamiltonian systems. In: {\it Dynamical Systems in Classical Mechanics\/},
227--238, Amer. Math. Soc. Transl. Ser. 2, {\bf 168}, Amer. Math. Soc., Providence, RI, 1995

\bibitem{Man_so_n} 
Manakov S.V. Note on the integration of Euler's equations of the 
 dynamics of an $n$-dimensional rigid body. 
{\it Funct. Anal. Appl.\/} {\bf 10} (1976), 328--329 

%\bibitem{MPS} Marsden, J.E., S.Pekarsky, and S.Shkoller. 
%Discrete Euler--Poincar\'e and Lie--Poisson equations. 
%{\it Nonlinearity} {\bf 12} (1999), 1647 -1662

\bibitem{MiFo} Mishchenko, A. S. and A. T. Fomenko. On the integration of the Euler equations on
semisimple Lie algebras. {\it Sov. Math. Dokl.\/}, {\bf 17}, (1976), 1591--1593.

\bibitem{Moser_Chern} Moser J. Geometry of quadrics and spectral theory.
{\it Chern Symposium.\/} Berkley 1979 (1980), 147--188

\bibitem{MosVes} Moser J., Veselov A.
Discrete versions of some classical integrable systems and factorization
of matrix polynomials. {\it Comm. Math. Phys.\/} {\bf 139} (1991), 217--243

\bibitem{R} Ratiu T. The motion of the free $n$-dimensional rigid body.
{\it Indiana U. Math.\/} J.,29 (1980), 609-627

\bibitem{S} Suris,  Yu.B. Integrable discretizations for lattice systems: 
local equations of motion and their hamiltonian properties", {\it Rev.Math.Phys.\/} {\bf 11}
(1999), no. 6, 727--822

\bibitem{SB} Suris, Yu.B.
The problem of integrable discretization: Hamiltonian approach. Progress in Mathematics, {\bf 219}, Birkh\"auser Verlag, 2003


\bibitem{Ves} Veselov, A.P. Integrable discrete-time systems and difference operators.
{\it Funct. An. and Appl.\/} {\bf 22} (1988), 83--94.

\bibitem{Whitt} Whittaker E.T. A treatise on analytical dynamics. 4-d ed. ,
 Cambridge Univ. Press, Cambridge 1960

\end{thebibliography}
\end{document}


nlin0505045.extracted_bib
\begin{thebibliography}{35}

\bibitem{AHP} Adams M.R., Harnad J., and Previato E.
  Isospectral Hamiltonian flows in finite and infinite dimensions. I. 
Generalized Moser and moment maps in loop algebras.
{\it Comm. Math. Phys.\/} {\bf 117} (1988), 451--500

 \bibitem{AHH_CMP} Adams M.R., Harnad J., and Hurtubise J.
   Darboux Coordinates and Liouville--Arnold Integration in
   Loop Algebras. {\it Comm. Math. Phys.\/} {\bf 155} (1993), 385--413

 \bibitem{AHH_Lett} Adams M.R., Harnad J., and Hurtubise J.
   Darboux Coordinates on Coadjoint Orbits of Lie Algebras.
   {\it Let. Math. Phys.\/} {\bf 40} (1997), 41--57

\bibitem{Bl_Cr} Bloch, A.M. and Crouch P. Optimal control and geodesic flows. 
{\it Systems Control Lett.\/} {\bf 28} , no. 2, (1996),  65-72.


\bibitem{BCMR} Bloch, A. M., P. Crouch, J. E. Marsden and T. S. Ratiu. 
The symmetric representation of the rigid body equations and their discretization. {\it Nonlinearity} {\bf 15} (2002), no. 4, 1309--1341

\bibitem{Bob_Lorb}  Bobenko A. I., Lorbeer B., and Suris Yu.
Integrable discretizations of the Euler top
{\it J. Math. Phys.\/} {\bf 39}, no. 12 (1998), 6668--6683

\bibitem{Sovr_Geom} Dubrovin B.A., Novikov S.P., Fomenko A.T. 
Modern Geometry. Springer--Verlag, 1989 

\bibitem{Fe_AMS}  Fedorov Yu. Integrable systems, Lax representations, and confocal
   quadrics. In: {\it Dynamical systems in classical mechanics\/}, 173--199,
   Amer. Math. Soc. Transl. Ser. 2, {\bf 168}, Amer. Math. Soc., Providence, RI,
   1995

\bibitem{Acta_bill} Fedorov Yu.
   Classical integrable systems and billiards related   to generalized Jacobians.  {\it Acta Appl. Math.\/}
   {\bf 55} 3 (1999) 251--301

\bibitem{Sabaudia} Fedorov Yu. Discrete versions of some algebraic integrable systems
 related to generalized Jacobians. {\it SIDE III---symmetries and integrability
of difference equations\/} (Sabaudia, 1998), 147--160,
CRM Proc. Lecture Notes {\bf 25}, Amer. Math. Soc., Providence, RI, 2000

\bibitem{FFrahm}  Frahm F. \"Uber gewisse Differentialgleichungen.
{\it Math. Ann.\/} {\bf 8} (1874), 35--44

\bibitem{Harnad1} Harnad J. Dual Isomonodromic Deformations and 
Moment Maps in Loop Algebras. {\it Comm. Math. Phys.\/} {\bf 166} (1994), 337--365

\bibitem{Jac_corps} Jacobi K.G.
Sur la rotation d'un corps. In: {\it Gesamelte Werke\/} {\bf 2} (1884), 139--172

\bibitem{Jap} Hirota, Ryogo; Kimura, Kinji. Discretization of the Euler top. 
{\it J. Phys. Soc. Japan\/} {\bf  69} (2000), no. 3, 627--630

\bibitem{Koz_vortex}  Kozlov V.V.  The vortex theory of the top.
{\it Vestn. Moskov. Univ., Ser. I. Mat. Mekh.\/} No. 4 (1990) 56--62  (Russian)

\bibitem{Koz_AMS} Kozlov V. V. Hydrodynamics of noncommutative integration of
Hamiltonian systems. In: {\it Dynamical Systems in Classical Mechanics\/},
227--238, Amer. Math. Soc. Transl. Ser. 2, {\bf 168}, Amer. Math. Soc., Providence, RI, 1995

\bibitem{Man_so_n} 
Manakov S.V. Note on the integration of Euler's equations of the 
 dynamics of an $n$-dimensional rigid body. 
{\it Funct. Anal. Appl.\/} {\bf 10} (1976), 328--329 

%\bibitem{MPS} Marsden, J.E., S.Pekarsky, and S.Shkoller. 
%Discrete Euler--Poincar\'e and Lie--Poisson equations. 
%{\it Nonlinearity} {\bf 12} (1999), 1647 -1662

\bibitem{MiFo} Mishchenko, A. S. and A. T. Fomenko. On the integration of the Euler equations on
semisimple Lie algebras. {\it Sov. Math. Dokl.\/}, {\bf 17}, (1976), 1591--1593.

\bibitem{Moser_Chern} Moser J. Geometry of quadrics and spectral theory.
{\it Chern Symposium.\/} Berkley 1979 (1980), 147--188

\bibitem{MosVes} Moser J., Veselov A.
Discrete versions of some classical integrable systems and factorization
of matrix polynomials. {\it Comm. Math. Phys.\/} {\bf 139} (1991), 217--243

\bibitem{R} Ratiu T. The motion of the free $n$-dimensional rigid body.
{\it Indiana U. Math.\/} J.,29 (1980), 609-627

\bibitem{S} Suris,  Yu.B. Integrable discretizations for lattice systems: 
local equations of motion and their hamiltonian properties", {\it Rev.Math.Phys.\/} {\bf 11}
(1999), no. 6, 727--822

\bibitem{SB} Suris, Yu.B.
The problem of integrable discretization: Hamiltonian approach. Progress in Mathematics, {\bf 219}, Birkh\"auser Verlag, 2003


\bibitem{Ves} Veselov, A.P. Integrable discrete-time systems and difference operators.
{\it Funct. An. and Appl.\/} {\bf 22} (1988), 83--94.

\bibitem{Whitt} Whittaker E.T. A treatise on analytical dynamics. 4-d ed. ,
 Cambridge Univ. Press, Cambridge 1960

\end{thebibliography}