Commutative subalgebras of three first-order symmetry operators and separation of variables in a wave equation

The problem of complete separation of variables in a wave equation in the four-dimensional Minkowski space-time is considered. Unlike the well-known series of Kalnins and Miller studies, the present study is based on the theorem on necessary and sufficient conditions for complete separation of variables in the Shapovalov nonparabolic equation. Nonequivalent complete sets of three first-order differential symmetry operators are constructed, the corresponding privileged coordinate systems are determined and complete separation of variables in the wave equation is realized