On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces

All homogeneous spaces $G/K$ ($G$ is a semisimple complex (compact) Lie group, $K$ a reductive subgroup) are enumerated for which arbitrary Hamiltonian flows on $T^*(G/K)$ with $G$-invariant Hamiltonians are integrable in the class of Noether integrals. It is proved that only for these spaces $G/K$ does the quasiregular representation of $G$ in the space of regular functions of the algebraic variety $G/K$ have a simple spectrum.
Bibliography: 21 titles.