Integrable Hamiltonian systems connected with graded Lie algebras

In this paper there is given a geometric scheme for constructing integrable Hamiltonian systems based on Lie groups, generalizing the construction of M. Adler. The operation of this scheme is considered for parabolic decompositions of semisimple Lie groups. Fundamental examples of integrable systems are connected with graded Lie algebras. Among them are the generalized periodic chains of Toda, multidimensional tops, and the motion of a point on various homogeneous spaces in a quadratic potential.