Coupling symmetries with Poisson structures

TL;DR

This paper investigates local normal forms of integrable systems on Poisson manifolds with symmetries, exploring conditions for splitting and providing examples where splitting fails, thus advancing understanding of symmetry and structure in Poisson geometry.

Contribution

It analyzes the existence of Weinstein's splitting theorem in symmetric settings and presents examples where such splitting does not occur, highlighting limitations of current normal form results.

Findings

Weinstein's splitting theorem may not hold with symmetries.

Examples show non-splitting of integrable systems with symmetries.

Conditions for splitting in symmetric integrable systems are characterized.

Abstract

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.