  In an infinite dimensional Hilbert space we consider a family of commuting
analytic vector fields vanishing at the origin and which are nonlinear
perturbations of some fundamental linear vector fields. We prove that one can
construct by the method of Poincar\'e normal form a local analytic coordinate
transformation near the origin transforming the family into a normal form. The
result applies to the KdV and NLS equations and to the Toda lattice with
periodic boundary conditions. One gets existence of Birkhoff coordinates in a
neighbourhood of the origin. The proof is obtained by directly estimating, in
an iterative way, the terms of the Poincar\'e normal form and of the
transformation to it, through a rapid convergence algorithm.
