On the density of Lyapunov unstable elliptic equilibria

We prove that any real-analytic Hamiltonian in five or more degrees-of-freedom, with a locally integrable non-degenerate elliptic equilibrium with indefinite quadratic part, can be perturbed within the real analytic category, while preserving the Birkhoff normal form at the equilibrium up to any arbitrary order, so that the equilibrium becomes Lyapunov unstable.