Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, $SO(2,1)$ Symmetry, and Magnetic Uniqueness

The hyperbolic potential $$ V(r)=-\fracα{(R^2-r^2)^2} $$ was identified by Olshanii as the negative-curvature counterpart of Newton's off-center circular-orbit problem. We first show that this form is not an isolated ansatz. For a planar radial natural Hamiltonian at fixed energy, if the Jacobi metr