On the Hamiltonian Structure of Normal Forms at Elliptic Equilibria of Reversible Vector Fields in $R^4$

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-semisimple 1 : 1 resonance 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.