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In this work we shall present a survey on problems and results on singularholomorphic foliations and Pfaff systems on complex manifolds assuming thatthese objects possess invariant analytic varieties. We will focus on recentresults which have been motivated by classical works of Darboux, Poincar\'e andPainlev\'e on the problem of algebraic integration of singular polynomialdifferential equations. We present results on Poincar\'e and Painlev\'e problemof bounding the degree and the genus of analytic varieties invariant byholomorphic foliations and Pfaff systems. We shall discuss the general ideas ofthe theory of integrability characterizing the existence of meromorphic firstintegrals for complex analytic Pfaff equations.

Analytic varieties invariant by foliations and Pfaff systems