Finite-dimensional representations of Lie algebras and completely integrable systems

A method is presented for constructing completely integrable dynamical systems. It employs the translates of functions on a finite-dimensional, $\operatorname{Ad}^*$-invariant subspace of the space of analytic functions on the dual space of the Lie algebra. The method is applied to the construction of completely integrable systems on real forms of Borel subalgebras of the exceptional Lie algebras. An algorithm is proposed for the calculation of the semi-invariants of the representation $\operatorname{Ad}^*$ of Borel subalgebras of the exceptional Lie algebras.
Bibliography: 12 titles.