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Let M be a two dimensional complex manifold, p ∈ M and 	 a germ of holomorphic foliation of M  at p. Let	 be a germ of an irreducible, possibly singular, curve at p in M which is a separatrix for 	 . We prove that if the Camacho-Sad-Suwa index Ind 	 then there exists another separatrix for 	 at p. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.

Holomorphic dynamics near germs of singular curves