We consider the reduced two-body problem with a central potential on the sphere S 2 and the hyperbolic plane H 2 . For two potentials different from the Newton and the oscillator ones we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.

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We consider the two-body problem with central interaction on two-point homogeneous spaces from the point of view of the invariant differential operators theory. The representation of the two-particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connection of different mass center definitions for these spaces to the obtained expression for Hamiltonian operator is studied.

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