  We study pseudo-Riemannian Einstein manifolds which are conformally
equivalent with a metric product of two pseudo-Riemannian manifolds.
Particularly interesting is the case where one of these manifolds is
1-dimensional and the case where the conformal factor depends on both manifolds
simultaneously. If both factors are at least 3-dimensional then the latter case
reduces to the product of two Einstein spaces, each of the special type
admitting a non-trivial conformal gradient field. These are completely
classified. If each factor is 2-dimensional, there is a special family of
examples of non-constant curvature (called extremal metrics by Calabi), where
in each factor the gradient of the Gaussian curvature is a conformal vector
field. Then the metric of the 2-manifold is a warped product where the warping
function is the first derivative of the Gaussian curvature. Moreover we find
explicit examples of Einstein warped products with a 1-dimensional fibre and
such with a 2-dimensional base. Therefore in the 4-dimensional case our Main
Theorem points towards a local classification of conformally Einstein products.
Finally we prove an assertion in the book by A.Besse on complete Einstein
warped products with a 2-dimensional base. All solutions can be explicitly
written in terms of integrals of elementary functions.
