We study the existence of periodic solutions in the neighbourhood of
symmetric (partially) elliptic equilibria in purely reversible Hamiltonian vector
fields. These are Hamiltonian vector fields with an involutory reversing symmetry
$R$. We contrast the cases where $R$ acts symplectically and anti-symplectically.
In case $R$ acts anti-symplectically, generically purely imaginary eigenvalues
are isolated, and the equilibrium is contained in a local two-dimensional invariant
manifold containing symmetric periodic solutions encircling the equilibrium
In case $R$ acts symplectically, generically purely imaginary eigenvalues
are doubly degenerate, and the equilibrium is contained in two two-dimensional invariant
manifolds containing nonsymmetric periodic solutions encircling
the equilibrium point. In addition, there exists a three-dimensional invariant
surface containing a two-parameter family of symmetric periodic solutions.