Local monomialization of a system of first integrals of Darboux type

Abstract

Given a real- or complex-analytic singular foliation $\theta$ with $n$ first integrals of meromorphic or Darboux type $(f_1,\dots ,f_n)$, we prove that there exists a local monomialization of the first integrals. In particular, if $\theta$ is generated by the $n$ first integrals, we prove the existence of a local reduction of singularities of $\theta$ to monomial singularities.