On bi-Hamiltonian structure of some superintegrable systems

TL;DR

This paper explores the bi-Hamiltonian structures of certain superintegrable systems on symplectic four-dimensional Lie groups, classifying them and constructing associated control matrices to understand their integrability.

Contribution

It provides a classification of bi-Hamiltonian structures on specific Lie groups and introduces control matrices for these systems, advancing the understanding of their integrability.

Findings

Classification of bi-Hamiltonian structures on symplectic four-dimensional Lie groups.

Construction of control matrices for the classified structures.

Enhanced understanding of superintegrable systems' geometric properties.

Abstract

We discuss bi-Hamiltonian structures for integrable and superintegrable Hamiltonian system on the list of symplectic four-dimensional real Lie groups are classified by G. Ovando. In addition, we creat corresponding control matrix for obtained bi-Hamiltonian structures.