Axially Symmetric, Spatially Homothetic Spacetimes
  We show that the existence of appropriate spatial homothetic Killing vectors is directly related to the separability of the metric functions for axially symmetric spacetimes. The density profile for such spacetimes is (spatially) arbitrary and admits any equation of state for the matter in the spacetime. When used for studying axisymmetric gravitational collapse, such solutions do not result in a locally naked singularity. 
Sanjay M. Wagh and Keshlan S. Govinder