ИСТИНА

We study integrable Hamiltonian systems with 2 degrees of freedom on regular compact isoenergy 3-manifolds. Such a system is given by a pair (B,F) of a closed 2-form B without zeros and a Bott function F (called the first integral) on a compact 3-manifold Q endowed with a volume form, such that d(FB)=0. A.Bolsinov and A.Fomenko constructed in 1995, under some additional assumptions, a complete set {I_k} of orbital invariants I_k:{(B,F)}---->R of such integrable systems. From this classification it turned out that any orbital invariant of generic integrable systems is ``trivial'', i.e. it can be expressed in terms of local extremes of rotation functions on one-parameter families of invariant tori. Bolsinov and Fomenko posed the question which of (nontrivial) orbital invariants are ``stable'' under integrable perturbations, i.e. can be continuousely extended to a neighbourhood of the given class of (non-generic) integrable systems (clearly, the trivial invariants are stable). A simplified question can be formulated as follows. Suppose that two (non-generic) integrable systems are given, that have the same topology of the foliation by invariant submanifolds (for simplicity, have the same first integral). Is it possible to make these systems orbitally equivalent via small integrable perturbations, or some orbital invariants exist that can serve as an obstruction to this scenario? In our talk, we answer the above questions. We show that the key role is played by the topology of the singular invariant fiber (i.e. of the integral submanifold different from torus), namely, by the following property. It is known that near a singular invariant fiber the dynamical system always admits a cross-section, which is a 2-manifold with boundary. The key property is the genus of the cross-section. If the genus equals 0 then there are no nontrivial continuous orbital invariants I:{(B,F)}--->R. However, if the genus is positive, such invariants can appear, and we show which of the orbital invariants have this property in some genus 1 cases. This work was supported by the Russian Science Foundation grant (project No.17-11-01303).