Differential complex associated to closed differential forms of nonconstant rank

In the present paper we construct a complex of sheaves associated to a closed differential form $\omega$. We study this complex in case $\omega$ is a) a closed 1-form vanishing at an embedded submanifold, b) a symplectic structure with Martinet singularities. In particular, we prove that, under additional conditions on $\omega$, this complex gives a fine resolution for the sheaf of infinitesimal automorphisms of $\omega$.