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

The orbits in the potentials ψ = − C z R − 2 + ζ ( R ) , ( ζ arbitrary) have integrals E , R 2 ϕ ̇ = h and h z ̇ + C ϕ = I . 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.