Constants of motion in stationary axisymmetric gravitational fields

The motion of test particles in stationary axisymmetric gravitational fields is generally nonintegrable unless a non-trivial constant of motion, in addition to energy and angular momentum along the symmetry axis, exists. The Carter constant in Kerr-de Sitter space-time is the only example known to date. Proposed astrophysical tests of the black hole no-hair theorem have often involved integrable gravitational fields more general than the Kerr family, but the existence of such fields has been a matter of debate. To elucidate this problem, we treat its Newtonian analogue by systematically searching for non-trivial constants of motion polynomial in the momenta and obtain two theorems. First, solving a set of quadratic integrability conditions, we establish the existence and uniqueness of the family of stationary axisymmetric potentials admitting a quadratic constant. As in Kerr-de Sitter space-time, the mass moments of this class satisfy a 'no-hair' recursion relation M2l +2 = a²M2l, and the constant is Noether related to a second-order Killing-Stäckel tensor. Second, solving a new set of quartic integrability conditions, we establish non-existence of quartic constants. Remarkably, a subset of these conditions is satisfied when the mass moments obey a generalized 'no-hair' recursion relation M_{2l +4} = (a² + b²)_{M2l +2}-a²b²M2l. The full set of quartic integrability conditions, however, cannot be satisfied non-trivially by any stationary axisymmetric vacuum potential.