Integrability of quantum dots

We determine the frequency ratios τ≡ωz/ωρ for which the Hamiltonian system with a potentialV=1r+12(ωρ2(x2+y2)+ωz2z2) is completely integrable. We relate this result to the existence of conformal Killing tensors of the associated Eisenhart metric on R1,4. Finally we show that trajectories of a particle moving under the influence of the potential V are not unparametrised geodesics of any Riemannian metric on R3.