Assume that we have a $1$ dimensional foliation of $\mathbb{R}^2$. Is there a global diffeomorphism of the plane which map all leaves of the foliation to curves with non zero curvature? One can consider the same question for $1$ dimensional foliation of $\mathbb{R}^n$ requring the leaves transform to Frenet curves.

A motivation for this question:

When I was thinking to the following question, I was thinking to the orthonormal frame $\gamma',\gamma''$ as a possible resolution to find a metric compatible to our vector field:

Limit cycles as closed geodesics (in negatively or positively curved space)