On natural Poisson bivectors on the sphere

TL;DR

This paper explores natural Poisson bivectors on the sphere, providing a framework within bi-Hamiltonian geometry to understand most known integrable systems in this setting.

Contribution

It introduces the concept of natural Poisson bivectors and applies it to unify the understanding of integrable systems on the sphere.

Findings

Most known integrable systems on the sphere fit within the natural Poisson bivectors framework

The approach simplifies the classification of integrable systems on the sphere

Bi-Hamiltonian geometry is effective for analyzing Poisson structures on the sphere

Abstract

We discuss the concept of natural Poisson bivectors, which allows us to consider the overwhelming majority of known integrable systems on the sphere in framework of bi-Hamiltonian geometry.