Lie symmetries for the orbital linearization of smooth planar vector fields around singular points ✩

This work is a generalization of the method proposed in [I.A. García, S. Maza, Linearization of analytic isochronous centers from a given commutator, J. Math. Anal. Appl. 339 (1) (2008) 740–745] of linearization of analytic isochronous centers from a given commutator. In this paper we propose a constructive procedure to get the change of variables that orbitally linearizes a smooth planar vector field on C2 around an elementary singular point (i.e., a singular point with associated eigenvalues λ,μ ∈ C satisfying λ = 0) or a nilpotent singular point from a given infinitesimal generator of a Lie symmetry.