A characterization of revolution quadrics by a system of partial differential equations

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in $\text{Rn}$: paraboloid, ellipsoid, one sheet of a two-sheeted hyperboloid, and a hyperplane.