Partially superintegrable systems on Poisson manifolds

TL;DR

This paper extends the concept of superintegrable systems to Poisson manifolds, relaxing traditional restrictions and generalizing the Mishchenko-Fomenko theorem for broader applicability.

Contribution

It introduces the notion of partially superintegrable systems on Poisson manifolds and formulates an extension of the Mishchenko-Fomenko theorem for these systems.

Findings

Defines partially superintegrable systems on Poisson manifolds

Extends the Mishchenko-Fomenko theorem to these systems

Provides a framework for generalized action-angle coordinates

Abstract

Superintegrable systems on a symplectic manifold conventionally are considered. However, their definition implies a rather restrictive condition 2n=k+m where 2n is a dimension of a symplectic manifold, k is a dimension of a pointwise Lie algebra of a superintegrable system, and m is its corank. To solve this problem, we aim to consider partially superintegrable systems on Poisson manifolds where k+m is the rank of a compatible Poisson structure. The according extensions of the Mishchenko-Fomenko theorem on generalized action-angle coordinates is formulated.