On a Trivial Family of Noncommutative Integrable Systems

We discuss trivial deformations of the canonical Poisson brackets associated with the Toda lattices, relativistic Toda lattices, Henon-Heiles, rational Calogero-Moser and Ruijsenaars-Schneider systems and apply one of these deformations to construct a new trivial family of noncommutative integrable systems

Superintegrable St\"ackel Systems on the Plane: Elliptic and Parabolic Coordinates

Recently we proposed a generic construction of the additional integrals of motion for the St\"ackel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for elliptic and parabolic coordinate systems on the plane

On Integrable Perturbations of Some Nonholonomic Systems

Integrable perturbations of the nonholonomic Suslov, Veselova, Chaplygin and Heisenberg problems are discussed in the framework of the classical Bertrand-Darboux method. We study the relations between the Bertrand-Darboux type equations, well studied in the holonomic case, with their nonholonomic counterparts and apply the results to the construction of nonholonomic integrable potentials from the known potentials in the holonomic case

On Classical r-Matrix for the Kowalevski Gyrostat on so(4)

We present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On the Darboux-Nijenhuis Variables for the Open Toda Lattice

We discuss two known constructions proposed by Moser and by Sklyanin of the Darboux-Nijenhuis coordinates for the open Toda lattice.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

New Variables of Separation for the Steklov-Lyapunov System

A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of separation on this phase space for the Steklov-Lyapunov system and it's gyrostatic deformation

Integrability of Nonholonomic Heisenberg Type Systems

We show that some modern geometric methods of Hamiltonian dynamics can be directly applied to the nonholonomic Heisenberg type systems. As an example we present characteristic Killing tensors, compatible Poisson brackets, Lax matrices and classical $r$-matrices for the conformally Hamiltonian vector fields obtained in a process of reduction of Hamiltonian vector fields by a nonholonomic constraint associated with the Heisenberg system