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

Unfolding of nilpotent equilibria of degree 4 in Hamiltonian systems with 2 degrees of freedom