Keeping track of limit cycles via certain second order differential operator

Question

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.

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 $\gamma_2$?

On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

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"

Answer 1

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 $y^2 + 2x^2$.