Purely radiative perfect fluids with a 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 σ is degenerate, H, E and σ necessarily commute and hence the resulting spacetimes are hypersurface homogeneous of Bianchi class A (modulo some purely electric exceptions).