A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

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

Hilbert 16th problem and dynamical Lefschetz trace formula

• The title recalls me " The Horrible and Terrifying Deeds and Words of the Very Renowned Pantagruel King of the Dipsodes, Son of the Great Giant Gargantua" (see en.wikipedia.org/wiki/Gargantua_and_Pantagruel). – user64494 Jun 13 at 18:29
• @user64494 what is the reason the title of this post recalls you the title of that novel/ – Ali Taghavi Jun 13 at 21:00