Can we classify all functions whose gradient is an eigenvector of the Hessian?

Question

Let's treat the case of two dimensions, then we don't have freedom with the other eigenvector. Is there any classification of all smooth functions $f$ on $\mathbb{R^2}$ such that $\nabla f$ is an eigenvector of $D^2f$ at every point? Such functions solve a second-order quasilinear PDE, but makes me wonder if there is a reference with a more "explicit" classification.