First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's equation (In Besse's notation) $$r - \frac{1}{2} s\, g = 0,$$ but something related with a well known generalization known as parallel Ricci condition, i.e. $Dr = 0$. Our condition is (in components) $$\nabla_{[a} R_{b]c} = 0.$$

As you noticed, I found some information about parallel Ricci manifolds in Arthur Besse book. However I'd like to know if you know recent works on the subject, specially since our condition is different.

I found this post where @vladimir-s-matveev and @robert-bryant discuss a related topic.

Comments, explanations, references (books, lecture notes, articles or other) are all welcome. Thank you.

PD: I forgot to tell that we're interested in general integrability (solutions) of parallel Ricci condition, not necessarily on Riemannian (or Lorentzian) manifolds... they could be affine spaces, or metric spaces without metricity condition ($\nabla g \neq 0$).