In this paper natural generalizations are developed of the theory of elliptic operators invariant under the action of a C*-algebra. The theory of compact and Fredholm operators acting in spaces of the type of a Hilbert space over a C*-algebra is developed. A formula of the Atiyah-Singer type for elliptic operators over a C*-algebra is developed. Bibliography: 16 titles.

In this paper, a special class of dynamical systems is studied-the so-called Euler equations (a natural generalization of the classical equations of motion of a rigid body with one fixed point). It turns out that for any finite-dimensional Lie algebra this system has a large collection of integrals which are in involution. For the class of semisimple Lie algebras and for certain series of solvable Lie algebras these integrals turn out to be sufficient for the complete integration (using Liouville's theorem) of the multiparametric family of Euler equations.Bibliography: 8 titles.