  The very long-term evolution of the hierarchical restricted three-body
problem with a slightly aligned precessing quadrupole potential is investigated
analytically for librating Kozai-Lidov cycles (KLCs). \citet{klein2023}
presented an analytic solution for the approximate dynamics on a very long
timescale developed in the neighborhood of the KLCs fixed point where the
eccentricity vector is close to unity and aligned (or anti aligned) with the
quadrupole axis and for a precession rate equal to the angular frequency of the
secular Kozai-Lidov Equations around this fixed point. In this Letter, we
generalize the analytic solution to encompass a wider range of precession
rates. We show that the analytic solution approximately describes the
quantitative dynamics for systems with librating KLCs for a wide range of
initial conditions, including values that are far from the fixed point which is
somewhat unexpected. In particular, using the analytic solution we map the
strikingly rich structures that arise for precession rates similar to the
Kozai-Lidov timescale (ratio of a few).
