Linearizability and integrability of vector fields via commutation

In this paper, we consider complex smooth and analytic vector fields X in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field Y such that the Lie brackets [X,Y] ≡ 0. For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field Y, such that Y is a normalizer of X, i.e., [X,Y] = μX.  2005 Elsevier Inc. All rights reserved.