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This paper defines a class of variational problems on Lie groups that admitinvolutive automorphisms. The maximum Principle of optimal control thenidentifies the appropriate left invariant Hamiltonians on the Lie algebra ofthe group. The corresponding Hamiltonian equations admit a spectralrepresentation with important applications to the theory of integrable systems.Particular cases make a link to Moser's work relating the solutions of Kepler'sproblem to the geodesics on spaces of constant curvature, Newmann's mechanicalproblem on the sphere and related geodesic problem of Jacobi on an ellipsoid.

Integrable Hamiltonian systems on symmetric spaces: Jacobi, Kepler and Moser