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For a fixed frequency vector $\omega \in \mathbb{R}^2 \, \setminus \, \lbrace0 \rbrace$ obeying $\omega_1 \omega_2 < 0$ we show the existence ofGevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorovnon-degenerate analytic Hamiltonian, having a Lyapunov unstable ellipticequilibrium with frequency $\omega$. In particular, the elliptic fixed pointsthus constructed will be KAM stable, i.e. accumulated by invariant tori whoseLebesgue density tend to one in the neighbourhood of the point and whosefrequencies cover a set of positive measure.  Similar examples for near-integrable Hamiltonians in action-angle coordinatesin the neighbourhood of a Lagragian invariant torus with arbitrary rotationvector are also given in this work.

Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom