  When a singular point of a vector field passes through resonance, a formal
invariant cone appears. In the seventies, Pyartli proved that for
$(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a
family of invariant cylinders. In his thesis, Stolovitch established a new type
of normal form and proved that for a simple resonance and under arithmetic
conditions the cone is (the germ of) an analytic variety. In this paper, we
prove a versal deformation theorem for analytic vector fields with an isolated
singularity over Cantor sets. Our result implies that, under arithmetic
conditions, the resonant cone is the degeneration of a set of invariant
manifolds like in Pyartli's example. For the multi-Hopf bifurcation, that is
for the $(-1,1)^d$-resonance, this implies the existence of vanishing tori
carrying quasi-periodic motions generalising previous results of Chenciner and
Li.
