Differential characterization of quadratic surfaces

Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $\Omega$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions.