On a Poincaré lemma for singular foliations and geometric quantization

In this paper we prove a Poincar\'e lemma for forms tangent to a foliation with nondegenerate singularities given by an integrable system on a symplectic manifold. As a consequence, the Kostant complex in Geometric Quantization is a fine resolution of the sheaf of flat sections when the polarization is spanned by the Hamiltonian vector fields of the first integrals of this integrable system.