Orbital linearization around equilibria of vector fields in RnRn and Lie symmetries

In this paper we study some relationships between analytic vector fields XX in RnRn which are analytically orbitally linearizable in a neighborhood of an equilibrium point and the structure of the set N(X)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 XX via the existence of special elements in N(X)N(X) are given.