  Some geometric solutions to the problem of endowing a dynamical system with a
Hamiltonian formulation are presented within the class of Poisson structures.
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, 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.
