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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'.

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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.

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.