Weak instability and isochrony for Hamiltonian systems

We show some integrable Hamiltonian systems in dimension 4 with linearly
stable equilibria which are Lyapunov unstable and do not have asymptotic  motions.
No explicit example of such weak instability seems to be known in the
literature.
Our systems belong to a class with some rare elements which have stable
equilibria and isochronous periodicity of all orbits, superintegrable
systems. Some of them can be obtained by means of Noether's variational theorem too.

Gaetano Zampieri, Completely integrable Hamiltonian systems with weak
Lyapunov instability or isochrony, to appear in Comm. Math. Phys.

Gianluca Gorni, Gaetano Zampieri, Variational Noether's theorem: the
interplay of time, space and gauge. In preparation.