Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown

Abstract

We develop a covariant Hamiltonian formulation of the Mathisson-Papapetrou-Tulczyjew-Dixon dynamics at quadratic order in spin under the Tulczyjew-Dixon spin supplementary condition (TD SSC). In four-dimensional, type-D Einstein (vacuum/$\Lambda$-vacuum) spacetimes admitting a non-degenerate Killing-Yano (KY) tensor, we reduce via a Dirac bracket to the 10-dimensional physical phase space and model the quadratic sector with a spin-induced quadrupole characterized by a deformability $\kappa$ ($\kappa=1$ for black-hole--like; $\kappa\neq 1$ for material or exotic compact objects). For $\kappa=1$, we construct five independent first integrals -- an autonomous Hamiltonian, two KY-generated Killing invariants, a linear R\"udiger constant, and a quadratic Carter-R\"udiger constant -- establishing Liouville-Arnold integrability at quadratic order in spin. For $\kappa\neq 1$, the symmetry-generated invariants are not conserved in general and integrability does not persist at this order. The proof proceeds via covariant Poisson-bracket computations using a null bivector decomposition; Kerr is recovered as a special case. These results show that integrability can persist beyond Kerr and beyond the linear-in-spin regime, laying groundwork for symmetry-based, beyond-Kerr modelling of asymmetric-mass, spinning compact binaries.