Integration of the GHP equations in spacetimes admitting a geodesic   shear-free expanding null congruence
  We perform a complete rho-integration of the GHP equations for all spacetimes that admit a geodesic shear-free expanding null congruence, whose Ricci spinor is aligned to the congruence and whose Ricci scalar is constant. We also deduce the system of GHP equations after the integration is completed, and discuss a few applications. 
Fredrik Andersson