ИСТИНА

We study singularities of the Lagrangian fibration given by a completely integrable system. A general problem of the theory of singularities of integrable systems is to describe the topology of singular fibers and their saturated neighborhoods (similarly for singular orbits). In this talk, we discuss just one particular type of singularitites, namely parabolic orbits and cuspidal tori (speaking informally, a cuspidal torus is a compact singular fiber that contains one parabolic orbit and no other singular points). We assume that the integrable system is real-analytic. An important property of parabolic orbits is their stability under small integrable perturbations (informally speaking, a singularity is called structurally stable if the topology of the fibration is preserved after any (small enough) real-analytic integrable perturbations of the system). This is one of the reasons why such orbits can be observed in many examples of integrable Hamiltonian systems (e.g. Kovalevskaya top and other integrable cases in rigid body dynamics). We give a classification, up to real-analytic symplectic equivalence, of the Lagrangian fibrations in a neighbourhood of a cuspidal torus (and similarly for a parabolic orbit).