Consider the second-order nonlinear PDE

$$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$

This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ of the form $\varphi(x,y) = \ell(x)\cdot r(y)$ (any $\Omega\subseteq\mathbb{R}^2$ open).

I'd like to know whether each (classical) solution of the PDE is of this form.

Are you aware of any 'uniqueness' results for the above PDE which might (dis)prove this?