Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems

TL;DR

This paper introduces a new family of integrable deformations for the Bogoyavlenskij-Itoh Lotka-Volterra systems, including a Lax operator with spectral parameter, expanding understanding of their integrability and connections to Veselov-Shabat systems.

Contribution

It constructs compatible Poisson structures and Casimir functions that generate integrals of motion for the deformed systems, providing a novel approach to their integrability.

Findings

Constructed a Lax operator with spectral parameter.

Established a family of compatible Poisson structures.

Linked deformations to Veselov-Shabat systems.

Abstract

We construct a family of integrable deformations of the Bogoyavlenskij-Itoh systems and construct a Lax operator with spectral parameter for it. Our approach is based on the construction of a family of compatible Poisson structures for the undeformed system, whose Casimirs are shown to yield a generating function for the integrals in involution of the deformed systems. We show how these deformations are related to the Veselov-Shabat systems.