Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

We study the simultaneous linearizability of d–actions (and the corresponding d-dimensional Lie algebras) defined by commuting singular vector fields in ℂ n fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of d vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the ℂ ∞ category, the situation is completely different. We show Sternberg’s theorem for a commuting system of ℂ ∞ vector fields with a Jordan block although they do not satisfy the condition