ИСТИНА

The talk is based on a joint work with A. Bolsinov and L. Guglielmi (2018). We study parabolic orbits with resonances (also known as Kalashnikovs typical rank-1 singularities) and the singular bres containing these orbits (known as cuspidal tori) for Lagrangian fibrations on symplectic 4-manifolds. An important property of parabolic orbits is their stability under small integrable perturbations (Kalashnikov 1998, Zung 2000). This is one of the reasons why such singularities can be observed in many examples of integrable Hamiltonian systems. It is well known that, from the smooth point of view, all parabolic orbits of a given resonance k:n are equivalent, i.e. any two parabolic orbits with the same resonance admit fibrewise diffeomorphic neighbourhoods. The same is true for cuspidal tori.We show that, in contrast to non-degenerate singularities (of elliptic, hyperbolic and focus-focus types), there exist parabolic orbits which are locally fibrewise diffeomorphic, but not symplectomorphic. Furthermore, all symplectic invariants of parabolic orbits (with a given resonance) can be expressed in terms of action variables. Finally, we show that the only symplectic semi-local invariant of a cuspidal torus (with a given resonance) is the canonical integer affine structure on the base of the corresponding singular Lagrangian fibration.