2026/01/10 by Paul Ramond, Soichiro Isoyama, Adrien Druart
#gr-qc #nlin.SI
We investigate the integrability of spinning compact body dynamics at quadratic order in spin in four-dimensional Einstein spacetimes admitting a non-degenerate Killing--Yano tensor. Working within the Mathisson--Papapetrou--Tulczyjew--Dixon framework under the Tulczyjew--Dixon spin supplementary condition, we model the spin-induced quadrupole with a deformability parameter κ, where κ=1 corresponds to black holes. The dynamics is formulated as a Hamiltonian system on a 10-dimensional physical phase space obtained by Dirac--Bergmann reduction. For κ=1, we establish Liouville--Arnold integrability at quadratic order in spin by constructing five independent, Poisson-commuting first integrals, including a generalization of the Carter constant and the Rüdiger constant to quadratic-in-spin order in Einstein spacetimes beyond Kerr. For κ≠ 1, the Rüdiger and Carter constants are no longer conserved; integrability does not persist at this order. All our results are carried out in a covariant manner and numerically verified, and Kerr is recovered as a special case. These results show that integrability can extend beyond Kerr and beyond the linear-in-spin regime, while its breakdown for κ≠ 1 points to the spin-induced quadrupole as a decisive probe of compact body structure.