Reversible Hamiltonian Liapunov center theorem

We study the existence of periodic solutions in the neighbourhood ofsymmetric (partially) elliptic equilibria in purely reversible Hamiltonian vectorfields. These are Hamiltonian vector fields with an involutory reversing symmetryR. We contrast the cases where R acts symplectically and anti-symplectically.In case R acts anti-symplectically, generically purely imaginary eigenvaluesare isolated, and the equilibrium is contained in a local two-dimensional invariantmanifold containing symmetric periodic solutions encircling the equilibriumpoint.In case R acts symplectically, generically purely imaginary eigenvaluesare doubly degenerate, and the equilibrium is contained in two two-dimensional invariantmanifolds containing nonsymmetric periodic solutions encirclingthe equilibrium point. In addition, there exists a three-dimensional invariantsurface containing a two-parameter family of symmetric periodic solutions.