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The determination of solutions of the Jacobi partial differential equations(PDEs) for finite-dimensional Poisson systems is considered. In particular, anovel procedure for the construction of solution families is developed. Such aprocedure is based on the use of time reparametrizations preserving theexistence of a Poisson structure. As a result, a method which is valid forarbitrary values of the dimension and the rank of the Poisson structure underconsideration is obtained. In this article two main families of timereprametrizations of this kind are characterized. In addition, these resultslead to a novel application which is also developed, namely the global andconstructive determination of the Darboux canonical form for Poisson systems ofarbitrary dimension and rank two, thus improving the local result provided byDarboux theorem for such a case.

Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems