Global resolution of singularities subordinated to a 1-dimensional foliation

Let M   be an analytic manifold over RR or CC, θ   a 1-dimensional Log-Canonical (resp. monomial) singular distribution and II a coherent ideal sheaf defined on M  . We prove the existence of a resolution of singularities for II that preserves the Log-Canonicity (resp. monomiality) of the singularities of θ. Furthermore, we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one.