  We provide a detailed derivation of the analytical expansion of the lunar and
solar disturbing functions. Although there exist several papers on this topic,
many derivations contain mistakes in the final expansion or rather (just) in
the proof, thereby necessitating a recasting and correction of the original
derivation. In this work, we provide a self-consistent and definite form of the
lunisolar expansion. We start with Kaula's expansion of the disturbing function
in terms of the equatorial elements of both the perturbed and perturbing
bodies. Then we give a detailed proof of Lane's expansion, in which the
elements of the Moon are referred to the ecliptic plane. Using this approach
the inclination of the Moon becomes nearly constant, while the argument of
perihelion, the longitude of the ascending node, and the mean anomaly vary
linearly with time.
  We make a comparison between the different expansions and we profit from such
discussion to point out some mistakes in the existing literature, which might
compromise the correctness of the results. As an application, we analyze the
long--term motion of the highly elliptical and critically--inclined Molniya
orbits subject to quadrupolar gravitational interactions. The analytical
expansions presented herein are very powerful with respect to dynamical studies
based on Cartesian equations, because they quickly allow for a more holistic
and intuitively understandable picture of the dynamics.
