Orthogonal and non-orthogonal separation of variables in the wave equation utt-uxx+V(x)u=0utt-uxx+V(x)u=0

We develop a direct approach to the separation of variables in partial differential equations. Within the framework of this approach, the problem of the separation of variables in the wave equation with time-independent potential reduces to solving an over-determined system of nonlinear differential equations. We have succeeded in constructing its general solution and, as a result, all potentials V(x) permitting variable separation have been found. For each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study. It should be noted that the above approach yields both orthogonal and non-orthogonal systems of coordinates.