Noncommutative integration method for linear partial differential equations. Functional algebras and dimensional reduction

The noncommutative integration method for linear partial differential equations [1] is applied for the generalized algebraic constructions, so called functional algebras ($F$-algebras), for which commutator of basis elements is nonlinear function of these elements. A nontrivial example is considered for the integration of Klein–Gordon equation in curved space which does not allow a separation of variables. Classification of quadratic algebras of special structure is performed. The dimensional reduction method is presented for noncommutative integrable equation. The reduced equation has the number of independent variables less than the primary one, and more complicated symmetry.