Purely radiative perfect fluids with degenerate shear tensor
  We consider non-rotating geodesic perfect fluid spacetimes which are purely radiative in the sense that the gravitational field satisfies the covariant transverse conditions div H = div E = 0. We show that when the shear tensor S is degenerate, H, E and S necessarily commute and hence the resulting spacetimes are hypersurface homogeneous of Bianchi class A (modulo some purely electric exceptions). 
H.R. Karimian, N. Van den Bergh and L. De Groote