  We determine the frequency ratios $\tau\equiv \omega_z/\omega_{\rho}$ for
which the Hamiltonian system with a potential \[
V=\frac{1}{r}+\frac{1}{2}\Big({\omega_{\rho}}^2(x^2+y^2)+{\omega_z}^2 z^2\Big)
\] is completely integrable. We relate this result to the existence of
conformal Killing tensors of the associated Eisenhart metric on $\mathbb{R}^{1,
4}$. 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 $\mathbb{R}^3$.
