Bihamiltonian geometry and separation of variables for Toda lattices

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifolds, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.Comment: 12 pages, Latex with amsmath and amssymb. Report of talks given at NEEDS9

Towards a classification of natural bi-hamiltonian systems

For construction and classification of the natural integrable systems we propose to use a criterion of separability in Darboux--Nijenhuis coordinates, which can be tested without an a priori explicit knowledge of these coordinates.Comment: LaTeX 22 page

On bi-integrable natural Hamiltonian systems on the Riemannian manifolds

We introduce the concept of natural Poisson bivectors, which generalizes the Benenti approach to construction of natural integrable systems on the Riemannian manifolds and allows us to consider almost the whole known zoo of integrable systems in framework of bi-hamiltonian geometry.Comment: 24 pages, LaTeX with AMSfonts (some new references were added

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

Dynamical R-Matrices for Integrable Maps

The integrability of two symplectic maps, that can be considered as discrete-time analogs of the Garnier and Neumann systems is established in the framework of the $r$-matrix approach, starting from their Lax representation. In contrast with the continuous case, the $r$-matrix for such discrete systems turns out to be of dynamical type; remarkably, the induced Poisson structure appears as a linear combination of compatible ``more elementary" Poisson structures. It is also shown that the Lax matrix naturally leads to define separation variables, whose discrete and continuous dynamics is investigated.Comment: 16 plain tex page