Hyperbolic Completion of Newton's Off-Center Orbit Problem: $SO(2,1)$ Symmetry, Inversion Duality, and Magnetic Classification

We resolve the hyperbolic off-center-orbit problem for the singular potential $$V(r)=-\frac{\alpha }{(R^2-r^2{)}^2},\alpha >0.$$ At zero energy, the Jacobi metric has constant negative curvature on both components separated by $r=R$. The interior is a constant multiple of the Poincaré disk metric, while circular inversion maps the exterior isometrically to the punctured disk. We classify all zero-energy trajectories: nonradial orbits are arcs of Euclidean circles orthogonal to $r=R$, radial trajectories lie on lines through the origin, and the force center lies outside every nonradial supporting circle. A Runge–Lenz-type moment map closes into $\mathfrak{s}\mathfrak{o}(2,1)$, whose Casimir equals the hyperbolic geodesic Hamiltonian. Canonical inversion preserves the symmetry generators and intertwines the zero-energy flows up to positive time reparametrization. In the magnetic problem, inversion becomes a charge-reversing $Q\leftrightarrow -Q$ duality preserving the shifted moment map and Casimir. The singular circle is reached in finite Newtonian time but lies at infinite Jacobi distance. Quantum mechanically, we distinguish the Stäckel coupling transform from genuine unitary equivalence and show that the bottom of the hyperbolic continuum maps to the Hardy/oscillation threshold of the inverse-square boundary model. Finally, the symmetry-preserving radial magnetic field becomes a constant intrinsic field on the hyperbolic plane. Its shifted Casimir classifies trajectories as magnetic circles, horocycles, or hypercycles, with a transition at $Q^2=8m\alpha R^2$. Numerical integrations confirm the orbit equations and conserved quantities.