  The aim of this paper is to introduce a class of Hamiltonian autonomous
systems in dimension 4 which are completely integrable and their dynamics is
described in all details. They have an equilibrium point which is stable for
some rare elements of the class, and unstable in most cases. Anyhow, it is
linearly stable (all orbits of the linearized system are bounded) and no motion
is asymptotic in the past, namely no non-constant solution has the equilibrium
as limit point as time goes to minus infinity. In the unstable cases, there is
a sequence of initial data which converges to the equilibrium point whose
corresponding solutions are unbounded and the motion is slow. So instability is
quite weak and perhaps no such explicit examples of instability are known in
the literature. The stable cases are also interesting since the level sets of
the 2 first integrals independent and in involution keep being non-compact and
stability is related to the isochronous periodicity of all orbits near the
equilibrium point and the existence of a further first integral. Hopefully,
these superintegrable Hamiltonian systems will deserve further research.
