An integrable family of Poisson systems: Characterization and global analysis

A family of solutions of the Jacobi PDEs is investigated. This family is defined for arbitrary values of the dimension n of the Poisson system; it is also of an arbitrary nonlinearity and can be globally analyzed (thus improving the usual local scope of the Darboux theorem). As an outcome of this analysis it is demonstrated that such Poisson structures lead to integrable systems. The solution family embraces as particular cases different systems of applied interest that are also regarded as examples.