Poincare normal form and carleman linearization: System of non linear ordinary differential equations

The goal of this paper is to present a generalization of the POINCARE - DULAC theorem about normal forms of vectors fields near an equilibrium point. We show that a derivation whose terms are formal power series in non commutative variables without constant terms can be transformed by a change of variables into a normal form whose “non - linear” terms are “resonnant”.