  On an orientable manifold M, we consider a regular even dimensional foliation
F which is globally defined by a set of k-independent 1-forms. We give
necessary and sufficient conditions for the existence of a regular Poisson
structure on M whose Characteristic foliation is precisely F. Moreover,
introducing a special class of the foliated 1-cohomology we describe
obstructions for the existence of unimodular Poisson structures with a given
characteristic foliation. In the same lines, we also give conditions for the
existence of transversally constant Poisson structures.
