  The very long-term evolution of the hierarchical restricted three-body
problem with a slightly aligned precessing quadrupole potential is studied
analytically. This problem describes the evolution of a star and a planet which
are perturbed either by a (circular and not too inclined) binary star system or
by one other star and a second more distant star, as well as a perturbation by
one distant star and the host galaxy or a compact-object binary system orbiting
a massive black hole in non-spherical nuclear star clusters
\citep{hamers2017,petrovich2017}. Previous numerical experiments have shown
that when the precession frequency is comparable to the Kozai-Lidov time scale,
long term evolution emerges that involves extremely high eccentricities with
potential applications for a broad scope of astrophysical phenomena including
systems with merging black holes, neutron stars or white dwarfs. By averaging
the secular equations of motion over the Kozai-Lidov Cycles (KLCs) we solve the
problem analytically in the neighborhood of the KLC fixed point where the
eccentricity vector is close to unity and aligned with the quadrupole axis and
for a precession rate similar to the Kozai Lidov time scale. In this regime the
dynamics is dominated by a resonance between the perturbation frequency and the
precession frequency of the eccentricity vector. While the quantitative
evolution of the system is not reproduced by the solution far away from this
fixed point, it sheds light on the qualitative behaviour.

High Energy Astrophysical Phenomena;Earth and Planetary Astrophysics;Solar and Stellar Astrophysics