Deterministic nature of conservative post-Newtonian accurate dynamics of compact binaries with leading order spin-orbit interaction

We formally show that the conservative second post-Newtonian (PN) accurate dynamics of spinning compact binaries moving in eccentric orbits, when spin effects are restricted to the leading order spin-orbit interaction cannot be chaotic for the following two distinct cases: (i) the binary consists of compact objects of arbitrary mass, where only one of them is spinning with an arbitrary spin and (ii) the binary consists of equal-mass compact objects, having two arbitrary spins. We rest our arguments on the recent determination of PN accurate Keplerian-type parametric solutions to the above cases, indicating that the PN accurate dynamics is integrable in these two situations. We compare predictions of our case (i) with those from a numerical investigation of an equivalent scenario that observed chaos in the associated dynamics. We also present possible reasons for the discrepancies.