Characterization, global analysis and integrability of a family of Poisson structures

TL;DR

This paper characterizes a family of Poisson structures by solving Jacobi equations, analyzing their global features, and proving integrability, with novel examples and formulations included.

Contribution

It provides a comprehensive analysis of a solution family of Jacobi equations, including Casimir invariants, Darboux form, and integrability proof, with new Poisson formulations.

Findings

Explicit characterization of the solution family of Jacobi equations.

Construction of Darboux canonical form for the structures.

Proof of integrability for the associated Poisson systems.

Abstract

An n-dimensional solution family of the Jacobi equations is characterized and investigated, including the global determination of its main features: the Casimir invariants, the construction of the Darboux canonical form and the proof of integrability for the related Poisson systems. Examples are given and include novel Poisson formulations.