  Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the
origin are characterised by the frequencies of the linearisation. Considering
the frequencies as parameters, the system undergoes a bifurcation when the
frequencies pass through a resonance. These bifurcations are well understood
for most resonances k:l, but not the semisimple cases 1:1 and 1:-1. A
two-degree-of-freedom Hamiltonian system can be approximated to any order by an
integrable approximation. The reason is that the normal form of a Hamiltonian
system has an additional integral due to the normal form symmetry. The latter
is intimately related to the ratio of the frequencies. Thus we study
$S^1$--symmetric systems. The question we wish to address is about the
co-dimension of such a system in 1:1 resonance with respect to
left-right-equivalence, where the right action is $S^1$--equivariant. The
result is a co-dimension five unfolding of the central singularity. Two of the
unfolding parameters are moduli and the remaining non-modal parameters are the
ones found in the linear unfolding of this system.
