Integrability of eccentric, spinning black hole binaries up to second post-Newtonian order

Accurate and efficient modeling of the dynamics binary black holes (BBHs) is crucial to their detection parameter estimation through gravitational waves, both with LIGO/Virgo LISA. General BBH configurations will have misaligned spins eccentric orbits, eccentricity being particularly relevant at early times. Modeling these systems analytically numerically challenging. Even though 1.5 post-Newtonian (PN) order Liouville integrable, numerical work has demonstrated chaos 2PN order, which impedes existence an analytic solution. In this article we revisit integrability 1.5PN orders. At 1.5PN, construct four (out five) action integrals. 2PN, show that system indeed integrable - but in a perturbative sense by explicitly constructing five mutually-commuting constants motion. Because KAM theorem, consistent past demonstration chaos. Our method extends higher PN orders, opening door for fully analytical solution generic eccentric, spinning problem.