  It is well known that pseudo-Riemannian metrics in the projective class of a
given torsion free affine connection can be obtained from (and are equivalent
to) the solutions of a certain overdetermined projectively invariant
differential equation. This equation is a special case of a so-called first BGG
equation. The general theory of such equations singles out a subclass of
so-called normal solutions. We prove that non-degerate normal solutions are
equivalent to pseudo-Riemannian Einstein metrics in the projective class and
observe that this connects to natural projective extensions of the Einstein
condition.
