Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems

TL;DR

This paper introduces a new family of Poisson structures in finite-dimensional systems, providing a global characterization, symplectic analysis, and generalizations of known solutions like separable structures.

Contribution

It presents a novel n-dimensional family of Poisson structures with comprehensive analysis and extends previous solutions such as separable Poisson structures.

Findings

Global characterization of the new Poisson family

Construction of symplectic and Darboux forms

Examples demonstrating generalizations of known structures

Abstract

A new n-dimensional family of Poisson structures is globally characterized and analyzed, including the construction of its main features: the symplectic structure and the reduction to the Darboux canonical form. Examples are given and include the generalization of previously known solution families such as the separable Poisson structures.