Abstract: Solutions of certain PDEs, such as Einstein metrics and soliton metrics, are easier to obtain if one assumes a certain amount of symmetry. We will discuss this in the case of polar actions and actions whose generic orbits are hypersurfaces. Smoothness conditions for solutions are an important part of the problem, which in general is known to be subtle and difficult to describe. We will discuss some recent progress by Tinyue Liu in the case of polar metrics and, in joint work with Luigi Verdiani, in the case of cohomogeneity one metrics.