On Weyl's Solution for Space-Times With Two Commuting Killing Fields

The solution of the metric coefficients for space-times with diagonal metrics and two commuting Killing fields can be reduced to a Laplace or a wave equation in two variables and to a further pair of integrable differential equations. This reduction can be achieved in a variety of ways. The choice of a coordinate frame and the selection of the combination of metric functions that satisfies the Laplace or the wave equation depend on the physical problem that is considered. The resolution of the issues that arise is illustrated in the contexts of three physical problems; and the solution of the remaining pair of equations, of most frequent occurrence in these contexts, is obtained in explicit form.