When is an axisymmetric potential separable?

Abstract

An axially symmetric potential psi(R,z)=psi(r,theta) is completely separable if the ratio s:k is constant. Here r*s=d^2(r^2*psi)/dr/d(theta) and k=d^2(psi)/dR/dz. If beta=s/k, then the potential admits an integral of the form of I=(L^2+beta*v_z^2)/2+xi where xi is some function of positions determined by the potential psi. More generally, an axially symmetric potential respects the third axisymmetric integral of motion -- in addition to the classical integrals of the Hamiltonian and the axial component of the angular momentum -- if there exist three real constants a,b,c (not all simultaneously zero, a^2+b^2+c^2>0) such that a*s+b*h+c*k=0 where r*h=d^2(r*psi)/d(sigma)/d(tau) and (sigma,tau) is the parabolic coordinate in the meridional plane such that sigma^2=r+z and tau^2=r-z.