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.
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