On the bi-Hamiltonian structure of Bogoyavlensky system on $so(4)$

TL;DR

This paper explores the bi-Hamiltonian structure of the Bogoyavlensky system on so(4), providing explicit methods for variable separation and analyzing its integrability properties.

Contribution

It introduces an explicit procedure for finding separation variables and relations in the bi-Hamiltonian Bogoyavlensky system on so(4).

Findings

Identified bi-Hamiltonian structure for the system.

Developed explicit separation of variables method.

Analyzed integrability with an additional fourth-order integral.

Abstract

We discuss bi-Hamiltonian structure for the Bogoyavlensky system on $so(4)$ with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation and the separation relations is considered in detail.