In the root-Kerr probe model, integrability holds to all spin orders at leading probe charge under Newman-Janis vertices but fails at spin-cubic order at second charge order and cannot be restored by further action deformation.
Generalized Carter & R\"udiger Constants of $\sqrt{\text{Kerr}}$
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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 $\sqrt{\text{Kerr}}$ field on flat spacetime: the electromagnetic field of a charged spinning ring-disk singularity obtained from the $G\to 0$ 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 R\"udiger'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.
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On the integrability of root-Kerr probe dynamics
In the root-Kerr probe model, integrability holds to all spin orders at leading probe charge under Newman-Janis vertices but fails at spin-cubic order at second charge order and cannot be restored by further action deformation.