2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/107570We consider the reduced two-body problem with the Newton and the oscillator potentials on the sphere ${\bf S}^{2}$ and the hyperbolic plane ${\bf H}^{2}$. For both types of interaction we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.20 pages, typos correctedDynamical SystemsMathematical Physics70F05; 37J30; 34M35; 70H07text