We prove that any real-analytic Hamiltonian with a locally integrable non-degenerate and not locally convex elliptic equilibrium, 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. This result is joint with Bassam Fayad, Jaime Paradela and Masha Saprykina.