A Riemannian manifold $(M,g)$ is called $\alpha-$Einstein if there exist a non-zero $1-$form $\alpha$ such $$\rho=ag+b\alpha\otimes\alpha$$ where $a,b$ are smooth functions on $M$ and $\rho$ is ricci tensor of $g$. It is easy to see that if $b=0$ then $(M,g)$ reduce to Einstein manifold.
Q1: Is there a similar version of the following known theorem such that we deduce $(M,g)$ is a $\alpha-$Einstein:
Theorem: Suppose $(M,g)$ is any Riemannian $n$-manifold with constant sectional curvature $C$. The curvature endomorphism, Ricci tensor, and scalar curvature of $g$ are given by the formulas $$R(X,Y)Z=C(g(Y,Z)X-g(X,Z)Y)$$ $$\rho=(n-1)Cg$$ $$r=n(n-1)C.$$
Q2: As you know Einstein metrics are fixed points of normalized ricci flow equation. Is there a PDE such as normalized ricci flow for $\alpha-$Einstein metrics?