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”