New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

We exhibit a concrete procedure to construct Einstein pseudo-K"ahler and para-K"ahler metrics on solvable Lie algebras. We apply this method to classify all the rank-one pseudo-Iwasawa extensions of type-(Nil4) nilsoliton in low dimension. We prove that such metrics exists on the rank-one pseudo-Iwasawa extension of the generalized Heisenberg Lie algebra. Further ideas and suggestions to produce more special Einstein pseudo-Riemannian metrics are exposed.