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We consider the existence of the periodic solutions in the neighbourhood of equilibria for C ∞ equivariant Hamiltonian vector fields. If the equivariant symmetry S acts antisymplectically and S 2 I, we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds each containing adoi:10.1155/2012/530209 fatcat:xhlw42c3w5hidko2namdmvglpm