A KAM phenomenon for singular holomorphic vector fields

Let X be a germ of holomorphic vector field at the origin of C and vanishing there. We assume that X is a “nondegenerate” good perturbation of a singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals. We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.