The anti-Einstein equations
  As we know, from the Einstein equations the vanishing of the four-divergence of the (symmetric) energy-momentum tensor follows. This is the case because the fourdivergence of the Einstein tensor (which is also symmetric) vanishes identically. Inverselly, we find that from the vanishing of the four-divergence of the energymomentum tensor (consisting in the general case of both symmetric and antisymmetric parts) not only the Einstein equations follow (concerning the symmetric part), but also the so-named anti-Einstein equations (concerning the antisymmetric part). These equations must be considered as complementary to the Einstein equations. Also, while from the Einstein equations the energy density (or the pressure) can be found (without using an equation of state), from the vanishing of the four-divergence of the (symmetric) energy-momentum tensor the pressure (or the energy density) can also be found, without having to use an additional (but arbitrary) equation of state. 
Evangelos Chaliasos