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We consider the existence of the periodic solutions in theneighbourhood of equilibria for ∞ equivariant Hamiltonian vector fields. If the equivariant symmetry  acts antisymplectically and 2=, we prove that genericallypurely imaginary eigenvalues are doubly degenerate and the equilibrium is containedin a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifoldseach containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltoniansystems

The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems