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Orbital invariants of integrable Hamiltonian systems. The case of simple systems. Orbital classification of systems of Euler type in rigid body dynamics

In this paper new orbital invariants of integrable Hamiltonian systems with two degrees of freedom are described, considered on non-singular three-dimensional constant-energy surfaces. A classification up to orbit-preserving homeomorphisms is obtained for dynamical systems that describe the rotation of a rigid body around its centre of mass for various values of the parameters.