We show that a partially massless graviton can propagate on a large set of
spacetimes which are not Einstein spacetimes. Starting from a recently
constructed theory for a massive graviton that propagates the correct number of
degrees of freedom on an arbitrary spacetime, we first give the full explicit
form of the scalar constraint responsible for the absence of a sixth degree of
freedom. We then spell out generic conditions for the constraint to be
identically satisfied, so that there is a scalar gauge symmetry which makes the
graviton partially massless. These simplify if one assumes that spacetime is
Ricci symmetric. Under this assumption, we find explicit non-Einstein
spacetimes (some, but not all, with vanishing Bach tensors) allowing for the
propagation of a partially massless graviton. These include in particular the
Einstein static Universe.