Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\mathcal{F}}$ be the Laplace operator on each leaf which comes by restriction of Riemannian metric of $M$ to leaves.
Let $A=\{f\in C^{\infty}(M)\mid \Delta_{\mathcal{F}}f=X.f\}$
Is $A$ invariant under the derivation $g\mapsto X.g$? Under which condition the later is the case? What is an example of this situation, invariance of $A$, with extra condition that the codimension of the derivation operator $(g\mapsto X.g)|A$ is a finite number?
This question is some how motivated by the following two posts(An indirect motivation not a direct motivation):
Elliptic operators corresponds to non vanishing vector fields
