Weakly Einstein conformal products

One says that a Riemannian four-manifold is \emph{weakly Einstein} if the three-index contraction of its curvature tensor against itself equals a function times the metric. Since this includes all four-manifolds that are Einstein, or conformally flat and scalar-flat, the term \emph{proper} may be used for weakly Einstein manifolds (or metrics) not belonging to the latter two classes. We establish two classification-type results about proper weakly Einstein metrics conformal to Riemannian products. This includes constructions of new examples, among them -- some of (local) cohomogeneity two, in contrast with the two previously known narrow classes of examples, having cohomogeneity zero and one. We also exhibit a simple coordinate description of one of the known examples, the EPS space, which shows that it is a conformal product and constitutes a single local-homothety type. Finally, we prove that there exist no proper weakly Einstein manifolds with harmonic curvature.