Inspired by the two posts which are linked bellow we ask the following question:


**Question:** For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ with $D(f)=(\Delta\circ L\_X-L\_X\circ \Delta)(f)$ where $\Delta$ is the standard Laplacian.



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> Is there a vector field $X$ on the plane with $2$ nested closed orbits $\gamma\_1 \subset \gamma\_2$ such that there exist a smooth function $f$ for which $D(f)$ does not vanish on the closur of the amnular region surrounded by $\gamma\_1$ and $\gamma\_2$?
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[On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold](https://mathoverflow.net/questions/334060/on-equation-delta-circ-partial-partial-x-partial-partial-x-circ-delta)


[Elliptic operators corresponds to non vanishing vector fields](https://mathoverflow.net/questions/182415/elliptic-operators-corresponds-to-non-vanishing-vector-fields)


**Remark:** The first words of the titles of this post is inspired by a paper by C.C. Pugh and J.P Francois " Keeping track of Limit cycles"



Keeping track of limit cycles via certain second order differential operator
I believe the answer is yes. 


Let $X$ be the vector field $2x \partial\_y - y \partial\_x$. The level sets of $y^2 + 2x^2$ are orbits of $X$, they have the shape of ellipses. 


It is easy to compute $D(f) = 2 \partial^2\_{xy} f$. So if you just let $f(x,y) = xy$ you in fact have $D(f) \equiv 2 \neq 0$, and in particular not on the annulus bounded by two level sets of $y^2 + 2x^2$. 


