On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in R4

Abstract This paper addresses the question whether normal forms of smooth reversible vector fields in R 4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between reversible and Hamiltonian normal forms. In the case of non-semi-simple 1 : 1 resonance and p : q resonance with p + q > 2 we establish a weaker form of equivalence, namely that of a formal orbital equivalence to a Hamiltonian normal form that involves an additional time-reparametrization of orbits. Moreover, in case p + q > 3 we show that no formal conjugacy to a Hamiltonian normal form exists.