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

The orbits in the potentials ψ=−CzR−2+ζ(R)ψ=−CzR−2+ζ(R), (ζζ arbitrary) have integrals EE, R2ϕ̇=h and hż+Cϕ=I. Thus the zz-velocity is proportional to the number of turns made around the axis! The Poisson Bracket [h,I][h,I] is not zero so Liouville’s integrability theorem does not apply. Starting from the self-similar potential with ζ∝R−1ζ∝R−1, we find some orbits that spiral around cones and explore general orbits in this strange system.