On integrability of Klein-Gordon equations in electromagnetic fields invariant under subgroups of the Poincare group

The structure of the symmetry operator algebra of the Klein-Gordon equation in an external electromagnetic field is described in terms of cohomologies of the Poincare subalgebras. An application of the method of noncommutative integration to constructing exact solutions of the Klein-Gordon equation is discussed. A classification of inequivalent electromagnetic fields invariant under simply transitive subgroups of the Poincare group, admitting the noncommutative integration of the Klein-Gordon equation, is presented.