  Accurate and efficient modeling of the dynamics of binary black holes (BBHs)
is crucial to their detection through gravitational waves (GWs), with
LIGO/Virgo/KAGRA, and LISA in the future. Solving the dynamics of a BBH system
with arbitrary parameters without simplifications (like orbit- or
precession-averaging) in closed form is one of the most challenging problems
for the GW community. One potential approach is using canonical perturbation
theory which constructs perturbed action-angle variables from the unperturbed
ones of an integrable Hamiltonian system. Having action-angle variables of the
integrable 1.5 post-Newtonian (PN) BBH system is therefore imperative. In this
paper, we continue the work initiated by two of us in arXiv:2012.06586, where
we presented four out of five actions of a BBH system with arbitrary
eccentricity, masses, and spins, at 1.5PN order. Here we compute the remaining
fifth action using a novel method of extending the phase space by introducing
unmeasurable phase space coordinates. We detail how to compute all the
frequencies, and sketch how to explicitly transform from the action-angle
variables to the usual positions and momenta. This analytically solves the
dynamics at 1.5PN. This lays the groundwork to analytically solve the
conservative dynamics of the BBH system with arbitrary masses, spins, and
eccentricity, at higher PN order, by using canonical perturbation theory.
