  We analyze the behavior of the argument of pericenter $\omega_2$ of an outer
particle in the elliptical restricted three-body problem, focusing on the
inverse Lidov-Kozai resonance. First, we calculate the contribution of the
terms of quadrupole, octupole, and hexadecapolar order of the secular
approximation of the potential to the outer particle's $\omega_2$ precession
rate $(d\omega_2/d\tau)$. Then, we derive analytical criteria that determine
the vanishing of the $\omega_2$ quadrupole precession rate
$(d\omega_2/d\tau)_{\text{quad}}$ for different values of the inner perturber's
eccentricity $e_1$. Finally, we use such analytical considerations and describe
the behavior of $\omega_2$ of outer particles extracted from N-body
simulations. Our analytical study indicates that the values of the inclination
$i_2$ and the ascending node longitude $\Omega_2$ associated with the outer
particle that vanish $(d\omega_2/d\tau)_{\text{quad}}$ strongly depend on the
eccentricity $e_1$ of the inner perturber. In fact, if $e_1 <$ 0.25 ($>$
0.40825), $(d\omega_2/d\tau)_{\text{quad}}$ is only vanished for particles
whose $\Omega_2$ circulates (librates). For $e_1$ between 0.25 and 0.40825,
$(d\omega_2/d\tau)_{\text{quad}}$ can be vanished for any particle for a
suitable selection of pairs ($\Omega_2$, $i_2$). Our analysis of the N-body
simulations shows that the inverse Lidov-Kozai resonance is possible for small,
moderate and high values of $e_1$. Moreover, such a resonance produces
distinctive features in the evolution of a particle in the ($\Omega_2$, $i_2$)
plane. In fact, if $\omega_2$ librates and $\Omega_2$ circulates, the extremes
of $i_2$ at $\Omega_2 =$ 90$^{\circ}$ and 270$^{\circ}$ do not reach the same
value, while if $\omega_2$ and $\Omega_2$ librate, the evolutionary trajectory
of the particle in the ($\Omega_2$, $i_2$) plane evidences an asymmetry respect
to $i_2 =$ 90$^{\circ}$.
