The generalized Einstein equations

Question

Let $(M,g)$ be an EInstein manifold, Ricci flat $Ric(g)=0$ and $X$ a vector field, I consider $M.{\bf R}$ and the metric $g_X$:

$$g_X = g +(X^*+dt) \otimes (X^* +dt)$$

The scalar curvature of $g_X$ is $r_X$. The generalized Einstein equations are:

$$X=(dr_X)^*$$

Have we non trivial solutions of the generalized Einstein equations?

Comments

What is the purpose of this question, i.e, why do you ask? Is there any physics background to this question, a physics-based motivation to attempt such a generalisation? Is  $X$ just any vector field? What is $t$? Probably it is related to "time", but is $(dt)^*$ just an arbitrary time-like vector or is there a specific choice involved about which your question is silent? In this context also note that your metric $g_X$ and  curvature $r_X$ are only labelled by $X$, any choice of $t$ does not show up. In what sense should $X=(dr_X)^*$ be a generalisation of the Einstein equations?

What if in $g_X$ you choose $X=-(dt)^*$?