  In the literature, the existence of Darboux polynomials and additional
polynomial first integrals has been considered in the case of Hamiltonian
systems. In this article such problem is formulated in the more general
framework of Poisson structures, which include Hamiltonian systems as a
particular case. This generalization allows a natural extension of the previous
results, which can now be applied to a larger class of vector fields and is
valid for arbitrary diffeomorphisms (instead of canonical transformations).
Examples are discussed.
