Integrable Euler equations associated with filtrations of Lie algebras

In semisimple Lie algebras a new construction is determined for symmetric operators such that the Euler equations reduce to chains of linear dynamical systems. The construction is associated with filtrations of Lie algebras and leads in a number of cases to completely integrable Euler equations. An analogous construction associated with filtrations of diffeomorphism groups is determined for Lie algebras of vector fields on manifolds. Constructions of Euler equations having sets of additional integrals in involution are found for the classical case.
Bibliography: 10 titles.