Unfamiliar integrals and motions down the `plug-hole' potential (Q1045867)

The orbits of a particle moving under the potential \(\psi = -CzR^{-2} + \zeta(R)\) are considered. This potential was earlier defined by one of the authors in \textit{D. Lynden-Bell} [Mon. Not. R. Astron. Soc. 124, 95--123 (1962; Zbl 0102.43801)]. The Poisson bracket \([h, I]\) is non-zero, so Liouville's integrability does not hold. The orbits have integrals \(E\), \(R^2 \dot{\phi}=h\) and \(h\dot{z} + C \phi = I\). The \(z\)-velocity is thus proportional to the number of turns made around the axis. Using a combination of numeric and analytical tools, the properties of the orbits for the cylindrical Keplerian problem, with \(\zeta = 2 \mu C R^{-1}\), are analysed here. At the end are some remarks concerning what one should expect for more general potentials \(\zeta\).