Commutative Poisson subalgebras for the Sklyanin bracket and deformations of known integrable models

TL;DR

This paper introduces a hierarchy of commutative Poisson subalgebras for the Sklyanin bracket, leading to new integrable models that deform classical systems like the Goryachev-Chaplygin top, Toda lattice, and Heisenberg model.

Contribution

It constructs a hierarchy of Poisson subalgebras for the Sklyanin bracket and derives new integrable models as deformations of classical systems.

Findings

Identified a hierarchy of commutative Poisson subalgebras.

Found integrable models as deformations of known systems.

Provided separation of variables for these models.

Abstract

A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the bracket, we find some integrable models and a separation of variables for them. The models obtained are deformations of known integrable systems like the Goryachev-Chaplygin top, the Toda lattice and the Heisenberg model.