We consider the motion of a charged spinning test/probe particle — governed by the Mathisson-Papapetrou-Dixon equations with generic, adiabatic, and conservative spin-and field-induced multipole moments — in a background \( \sqrt{\textrm{Kerr}} \) field on flat spacetime: the electromagnetic field of a charged spinning ring-disk singularity obtained from the G → 0 limit of the Kerr-Newman solution for a charged spinning black hole. We investigate the existence of two extra hidden constants of motion, analogous to the Carter constant (for geodesic motion in a Kerr spacetime, or for its spinning-probe generalization) and Rüdiger’s linear-in-spin constant for a spinning probe in a Kerr background. We find that these two constants exist only when the Wilson coefficients parameterizing the probe’s multipole structure take the particular values corresponding to “spin-exponentiation” of the effective Compton amplitudes through second order in spin.
