Anomaly Reparametrization of the Ligon--Schaaf Regularization in the Kepler problem

We revisit the Ligon--Schaaf regularization of the Kepler problem and identify the geometric origin of the rotation appearing in their transformation. We show that this rotation is determined by the eccentric anomaly of the Kepler motion, providing a transparent dynamical interpretation of the angle that renders the Kepler flow uniform on $T^{*}S^{3}$. Building on this insight, we extend the construction to positive and zero energies via the corresponding hyperbolic and parabolic anomalies, obtaining a unified geometric description of the Kepler flow across all energy levels.