I've been looking for an interesting problem, that is to find a quadratic form $F:\Bbb{R}^3 \to \Bbb{R}$, given by $F(x,y,z)=ax^2+by^2+cz^2 + dxy + exz + fyz + h$ such that image inverse of $F$ by $0$ is a regular surface with constant mean curvature. In this context, $0$ is regular value of $F$.
I appreciate any attempt to help