A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:

The differential operator $D(f)=\partial f/\partial X$ keeps invariant the space of Schwartz functions $\mathcal{S}$ and the differential operator $\mathcal{F}^{-1}D\mathcal{F}$ represents an ellptic operator on whole $\mathbb{R}^2$

What is an example of an elliptic foliation of the plane?

This question is inspired by the following post

A Fourier elliptic vector field on a Riemannian manifold

A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:

The differential operator $D(f)=\partial f/\partial X$ keeps invariant the space of Schwartz functions $\mathcal{S}$ and the differential operator $\mathcal{F}^{-1}D\mathcal{F}$ represents an ellptic operator on whole $\mathbb{R}^2$ where $\mathcal{F}$ is the Fourier transform defined on the space of Schwartz functions.

What is an example of an elliptic foliation of the plane?

This question is inspired by the following post

A Fourier elliptic vector field on a Riemannian manifold