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We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Let M be a smooth closed orientable surface, and let F be the space of Morse functions on M such that at least χ(M ) + 1 critical points of each function of F are labeled by different labels (enumerated). Endow the space F with C ∞ -topology. We prove the homotopy equivalence F ∼ R × M where R is one of the manifolds RP 3 , S 1 × S 1 and the point in dependence on the sign of χ(M ), and M is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space F are obtained.

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Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M having fixed number of critical points of each index, moreover at least χ(M ) + 1 critical points are labeled by different labels (enumerated). A notion of a skew cylindric-polyhedral complex, which generalizes the notion of a polyhedral complex, is introduced. The skew cylindric-polyhedral complex K (the "complex of framed Morse functions"), associated with the space F , is defined. In the case when M = S 2 , the polyhedron K is finite; its Euler characteristic χ( K) is evaluated and the Morse inequalities for its Betti numbers β j ( K) are obtained. A relation between the homotopy types of the polyhedron K and the space F of Morse functions, endowed with the C ∞ -topology, is indicated.

Обобщение теоремы Бертрана на поверхности вращения В работе доказано обобщение теоремы Бертрана на случай абстракт-ных поверхностей вращения, не имеющих "экваторов". Доказан критерий существования на такой поверхности ровно двух центральных потенциа-лов (с точностью до аддитивной и мультипликативной констант), для ко-торых все ограниченные орбиты замкнуты и имеется ограниченная неосо-бая некруговая орбита. Доказан критерий существования ровно одного такого потенциала. Изучены геометрия и классификация соответствую-щих поверхностей, с указанием пары (гравитационного и осцилляторного) потенциалов или единственного (осцилляторного) потенциала. Показано, что на поверхностях, не относящихся ни к одному из описанных классов, потенциалов искомого вида не существует.Ключевые слова: теорема Бертрана, обратная задача динамики, по-верхность вращения, движение в центральном поле, замкнутые орбиты D. A. Fedoseev, E. A. Kudryavtseva, O. A. Zagryadsky Generalization of Bertrand's theorem to surfaces of revolutionThe generalization of Bertrand's theorem to abstract surfaces of revolution without "equators" is proved. We prove a criterion for the existence on such a surface of exactly two central potentials (up to an additive and a multiplicative constants) all of whose bounded nonsingular orbits are closed and which admit a bounded nonsingular noncircular orbit. A criterion for the existence of a unique such potential is proved. The geometry and classification of the corresponding surfaces are described, with indicating either the pair of (gravitational and oscillator) potentials or the unique (oscillator) potential. The absence of the required potentials on any surface which does not meet the above criteria is proved.

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