  We study the existence of families of periodic solutions in a neighbourhood
of a symmetric equilibrium point in two classes of Hamiltonian systems with
involutory symmetries. In both classes, involutions reverse the sign of the
Hamiltonian function. In the first class we study a Hamiltonian system with a
reversing involution R acting symplectically. We first recover a result of
Buzzi and Lamb showing that the equilibrium point is contained in a three
dimensional conical subspace which consists of a two parameter family of
periodic solutions with symmetry R and there may or may not exist two families
of non-symmetric periodic solutions, depending on the coefficients of the
Hamiltonian. In the second problem we study an equivariant Hamiltonian system
with a symmetry S that acts anti-symplectically. Generically, there is no
S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we
prove the existence of at least 2 and at most 12 families of non-symmetric
periodic solutions. We conclude with a brief study of systems with both forms
of symmetry, showing they have very similar structure to the system with
symmetry R.
