  We consider Hamiltonian systems of two degrees of freedome having a nilpotent
equilibrium point with only one eigenvector. We provide the universal unfolding
of such equilibrium, provided a non-degeneracy condition holds. We show that
the only co-dimension 1 bifurcations that happen in the unfolding are of two
types: the normally hyperbolic or elliptic centre-saddle bifurcations and the
supercritical Hamiltonian-Hopf bifurcation.
