  For a fixed frequency vector $\omega \in \mathbb{R}^2 \, \setminus \, \lbrace
0 \rbrace$ obeying $\omega_1 \omega_2 < 0$ we show the existence of
Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov
non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic
equilibrium with frequency $\omega$. In particular, the elliptic fixed points
thus constructed will be KAM stable, i.e. accumulated by invariant tori whose
Lebesgue density tend to one in the neighbourhood of the point and whose
frequencies cover a set of positive measure.
  Similar examples for near-integrable Hamiltonians in action-angle coordinates
in the neighbourhood of a Lagragian invariant torus with arbitrary rotation
vector are also given in this work.
