Generalized Einstein equations

Question

Let $(M,g)$ be a riemannian manifold with Ricci curvature $Ric(g)$. The generalized Einstein equations are:

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

where $\Delta (g)$ is the Laplacian of the metric $g$, $\lambda ,\lambda'$ are constants.

Can we find black holes solutions of the generalized Einstein equations?

Comments

We could also take:

$$Ric(X,Y)=\lambda g(X,Y) + \lambda' [g(\nabla_X (dr^*),Y)+g(X,\nabla_Y (dr^*))]$$

where $r$ is the scalar curvature and $\nabla$ is the Levi-Civita connection.

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This is not an answer.

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What do you mean by the Laplacian of the metric?