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

https://mathoverflow.net/questions/356784/a-fourier-elliptic-vector-field-on-a-riemannian-manifold