  We study groups of germs of complex diffeomorphisms having a property called
irreducibility. The notion is motivated by the similar property of the
fundamental group of the complement of na irreducible hypersurface in the
complex projective space. Natural examples of such groups of germ maps are
given by holonomy groups and monodromy groups of integrable systems
(foliations) under certain conditions. We prove some finiteness results for
these groups extending previous results in [CL]. Applications are given to the
framework of germs of holomorphic foliations. We prove the existence of first
integrals under certain irreducibility or more general conditions on the
tangent cone of the foliation after a punctual blow-up.
