Generalized normal forms for polynomial vector fields☆☆This work was partially supported by two Projects of Ministerio de Educación y Ciencia (Spain), ESP99-1074-C02-01 and PB98-1576.

AbstractA method to obtain formal symmetries of polynomial vector fields with non-null linear part is presented. We show that, under certain conditions, the symmetries of the linear components can be extended to higher degree terms by means of adequate changes of variables. The approach is based on a generalization of the Normal Form Theorem for vector fields. Lie transformations for ordinary differential equations are used to extend the symmetries to any order. Reduced phase spaces are constructed by making use of the first integrals associated to the linear part of the new symmetry. For Hamiltonian vector fields, as the formal symmetries become integrals of motion, this procedure produces a reduction of the number of degrees of freedom at least by one. We illustrate the technique with some examples for Hamiltonian and non-Hamiltonian vector fields