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

Local monomialization of a system of first integrals of Darboux type