Nonlinear Poisson bracket,F-algebras,and noncommutative integration of linear differential equations

We study some properties of the nonlinear Poisson bracket and its analog for linear differential equations in partial derivatives (so-called F-algebras). We propose a method for noncommutative integration of linear differential equations for the case when the equation operator is embedded in an F-algebra. The method is based on the exact infinite irreducible representation of an F-algebra (-representation), which is introduced in the present paper.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 92–98, July, 1992.