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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?

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 $X=-(dt)^*$?

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