Unfamiliar integrals and motions down the ‘plug-hole’ potential

The orbits in the potentials ψ=−CzR⁻²+ζ(R), (ζ arbitrary) have integrals E, and . Thus the z-velocity is proportional to the number of turns made around the axis! The Poisson Bracket h,I] is not zero so Liouville’s integrability theorem does not apply. Starting from the self-similar potential with ζ∝R⁻¹, we find some orbits that spiral around cones and explore general orbits in this strange system.