Convergence of Poincare-Dulac normal form for commuting vector fields

We show that any m-tuple of linearly independent pairwise commuting holomorphic vector fields in a neighborhood of 0 in Cn with (n − m) common holomorphic first integrals (n ≥ m ≥ 1) admits a simultaneous convergent Poincaré-Dulac normalization. This result follows from another result which says that the existence of a convergent normalization for a vector field is equivalent to the existence of a local torus action which preserves the vector field. A similar result for the case of isochore vector fields is also obtained.