A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group $G$. Usually one consider a co-compact lactice $\Gamma$ on $G$ and the solvmanifold is $G/\Gamma$. Moreover, if there are left-invariant structures on $G$ (e.g. metrics, complex strucutres etc.), those structures define analogous structure on the compact quotient. Moreover, left-invariant structures are viewed as tensors on the Lie algebra $\mathfrak{g}$ of $G$.
I would like to know if there are known examples of solvmanifold with pseudo-Riemannian (or semi-Riemannian or indefinite) Einstein non-Ricci-flat metrics.
By the previous discussion, this question can be rephrase as follows: Are there examples of solvable Lie algebras admitting indefinite metrics (i.e. indefinite non-degenerate bilinear form) which are Einstein non-ricci-flat? Do those examples admit compact quotients?
Finally, remind that a nilpotent Lie group is a solvable Lie group, and a result of Maltcev assure that nilpotent Lie group with rational constant structures admits a co-compact lactice (hence a compact quotient).
