  Let $X$ be a germ of holomorphic vector field at the origin of ${\bf C}^n$
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.
