Orbital linearization around equilibria of vector fields in and Lie symmetries

In this paper we study some relationships between analytic vector fields X in R n which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set N ( X ) of its locally analytic normalizers (analytic infinitesimal generators of Lie point orbitally symmetries in that neighborhood). Characterization and necessary and sufficient conditions for the orbital linearization of X via the existence of special elements in N ( X ) are given.