Lowering the dimension of polynomial vector fields in R2 and R3

We classify the normal forms associated to polynomial vector fields with dimensions two and three whose principal part is linear. Then we reduce by one the dimension of the associated differential systems. This is achieved by means of the extension of a symmetry of the unperturbed part to the whole system, up to a certain order of approximation. The corresponding transformation is formal in the sense that we have no estimation on its radius of convergence. We calculate the invariants, reduced phase spaces and differential systems associated to each reduction. This is the general setting for the study of the dynamics of the reduced system in order to obtain information about the original one. (C) 2001 American Institute of Physics.