For almost every direction on a translation surface, the invariant surface of the flat geodesic flow persists within a finite-codimension family of smooth Hamiltonian perturbations.
Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Inventiones mathematicae 205 (3) (2016), 617--670
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Finite codimension stability of invariant surfaces
For almost every direction on a translation surface, the invariant surface of the flat geodesic flow persists within a finite-codimension family of smooth Hamiltonian perturbations.