We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}$ is a solution to the homogeneous Schrodinger equation with initial data $f$; that on cones (change $|\xi|^2$ to $|\xi|$) a solution to the wave equation with the same initial data; and on (part of) a spheres (change $|\xi|^2$ to $\sqrt{1-|\xi|^2}$) a solution to the Hamiltonian equation.

Is every extension operator on some smooth hypersurface (say, quadratic surfaces like hyperbolic hyperboloids, hyperbolic paraboloids, or more general surfaces that are not graphs of any quadratic or even polynomial functions) a solution to some PDE?