  The determination of solutions of the Jacobi partial differential equations
(PDEs) for finite-dimensional Poisson systems is considered. In particular, a
novel procedure for the construction of solution families is developed. Such a
procedure is based on the use of time reparametrizations preserving the
existence of a Poisson structure. As a result, a method which is valid for
arbitrary values of the dimension and the rank of the Poisson structure under
consideration is obtained. In this article two main families of time
reprametrizations of this kind are characterized. In addition, these results
lead to a novel application which is also developed, namely the global and
constructive determination of the Darboux canonical form for Poisson systems of
arbitrary dimension and rank two, thus improving the local result provided by
Darboux theorem for such a case.
