  Some positive answers to the problem of endowing a dynamical system with a
Hamiltonian formulation are presented within the class of Poisson structures in
a geometric framework. We address this problem on orientable manifolds and by
using decomposable Poisson structures. In the first case, the existence of a
Hamiltonian formulation is ensured under the vanishing of some topological
obstructions, improving a result of Gao. In the second case, we apply a variant
of the Hojman construction to solve the problem for vector fields admitting a
transversally invariant metric and, in particular, for infinitesimal generators
of proper actions. Finally, we also consider the hamiltonization problem for
Lie group actions and give solutions in the particular case in which the acting
Lie group is a low-dimensional torus.
