The coupled Einstein equations

Question

Let $M$ be a manifold with two riemannian metrics $(g,g')$. The coupled Einstein equations are:

$$Ric(g)= \lambda g'$$

$$Ric(g')=\lambda' g$$

with $Ric(g),Ric(g')$ the two Ricci curvatures and $\lambda,\lambda'$ two scalars.

Can these equations have physical meaning like a world with two leaves?