Characterization and global analysis of a family of Poisson structures

TL;DR

This paper characterizes a broad family of Poisson structures in three dimensions, providing explicit descriptions of their symplectic forms and Darboux canonical forms, with illustrative examples.

Contribution

It offers the first explicit global characterization of a three-dimensional family of Poisson structures, including their symplectic and Darboux forms.

Findings

Explicit global determination of symplectic structures

Construction of Darboux canonical forms

Examples illustrating the theoretical results

Abstract

A three-dimensional family of solutions of the Jacobi equations for Poisson systems is characterized. In spite of its general form it is possible the explicit and global determination of its main features, such as the symplectic structure and the construction of the Darboux canonical form. Examples are given.