Abstract
Time-dependent solutions of Einstein's vacuum field equations for axially symmetric non-rotating sources can be constructed exactly from key functions, which themselves are solutions of a certain key partial differential equation in two independent variables. Special examples of key functions (along with their consequential solutions of the field equations) are readily obtainable, but key functions describing physical models can only be obtained by solving a general problem.
The most promising approach for solving this general problem is to dispense with the parameter that appears in the key equation, thus enabling this third-order equation to be integrated to a second-order one, and introducing thereby an explicit arbitrary function into the equation. This function will need to be chosen in a model dependent manner.