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 were @vladimir-s-matveev and @robert-bryant discuss a related topic.

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