A family of warped product semi-Riemannian Einstein metrics

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an translation group and the fiber F is Ricci flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.