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

The problem of complex separation of variables in the wave equation is considered in four-dimensional Minkowskii space-time. In contrast to the known series of researches by Kalnins and Miller (see Ref. Zh., Fiz., 2B9 (1978); 1B208 and 1B209 (1979), e.g.), underlying this research is a theorem on the necessary and sufficient conditions of total separation of variables in the non-parabolic V. N. Shapovalov equation (Differents. Uravn., 16, No. 10, 1864-1874 (1980)). Nonequivalent complete sets of three differential first-order symmetry operators are constructed, appropriate coordinate systems are found, and complete separation of variables is performed in the wave equation.