POINCARE NORMAL FORMS AND LIE POINT SYMMETRIES

We study Poincare normal forms of vector fields in the presence of sym metry under general-i.e. not necessarily linear-diffeomorphisms. We sh ow that it is possible to reduce bath the vector field and the symmetr y diffeomorphism to normal form by means of an algorithmic procedure s imilar to the usual one for Poincare normal forms without symmetry; th is 'joint' normal form can be given a simple geometric characterizatio n.