Nature of the phase-flow pole

Determine whether the pole of the autonomous phase flow in the conformally flat isotropic pure Lovelock system corresponds to a curvature singularity or merely to a breakdown of the areal radial coordinate.

Background

The autonomous equation for the spatial metric potential has a pole at $Z^Q_\infty=-2k/(2k-1)$. The paper desingularises this pole by reparametrising the flow and shows that it becomes an ordinary point of the polynomially reparametrised dynamics.

However, the pole lies outside the physical Lorentzian static sector and outside the physical interval explored by the admissible solutions. Consequently, the phase-space analysis does not resolve whether the pole represents an actual curvature pathology or only a failure of the chosen areal-radius coordinate.