Carter-like Constants of the Motion in Newtonian Gravity and Electrodynamics

For a test body orbiting an axisymmetric body in Newtonian gravitational theory with mass $m$ and multiple moments $Q_{\ell }$ (and for a charge in orbit about a charge distribution with the same multipole moments) we show that there exists, in addition to the energy and angular momentum component along the symmetry axis, a conserved quantity analogous to the Carter constant of Kerr spacetimes for rotating black holes in general relativity, if the odd-$\ell$ moments vanish, and the even-$\ell$ moments satisfy $Q_{2\ell }=m(Q_2/m{)}^{\ell }$. Strangely, this is precisely the relation among mass moments enforced by the no-hair theorems of rotating black holes. By contrast, if Newtonian gravity is supplemented by a multipolar gravitomagnetic field, whose leading term represents frame dragging, we are unable to find an analogous Carter-like constant. This further highlights the special nature of the Kerr geometry.