On reductions on groups of geodesic flows with (left-) right-invariant metrics and their fields of symmetry

Dynamical systems in fluid mechanics, oceanography and meteorology (37N10) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics (37K65) Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics (70H33) Integrable cases of motion in rigid body dynamics (70E40)