Separable bi-Hamiltonian systems with quadratic in momenta first integrals

TL;DR

This paper reviews and advances the geometric theory of bi-Hamiltonian systems on Riemannian manifolds, introducing new classes of separable systems related to special conformal Killing tensors and their bi-Hamiltonian structures.

Contribution

It develops the separability theory for Gel'fand-Zakharevich bi-Hamiltonian systems and constructs infinitely many new separable systems via deformations of Benenti systems.

Findings

New classes of separable systems constructed

Systems can be lifted to bi-Hamiltonian form

Enhanced understanding of conformal Killing tensors

Abstract

Geometric separability theory of Gel'fand-Zakharevich bi-Hamiltonian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the separability of systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of separable systems are constructed by appropriate deformations of Benenti class systems. All such systems can be lifted to the Gel'fand-Zakharevich bi-Hamiltonian form.