  The behavior of an interior test particle in the secular 3-body problem has
been studied extensively. A well-known feature is the Lidov-Kozai resonance in
which the test particle's argument of periapse librates about $\pm 90^\circ$
and large oscillations in eccentricity and inclination are possible. Less
explored is the inverse problem: the dynamics of an exterior test particle and
an interior perturber. We survey numerically the inverse secular problem,
expanding the potential to hexadecapolar order and correcting an error in the
published expansion. Four secular resonances are uncovered that persist in full
$N$-body treatments (in what follows, $\varpi$ and $\Omega$ are the longitudes
of periapse and of ascending node, $\omega$ is the argument of periapse, and
subscripts 1 and 2 refer to the inner perturber and outer test particle): (i)
an orbit-flipping quadrupole resonance requiring a non-zero perturber
eccentricity $e_1$, in which $\Omega_2-\varpi_1$ librates about $\pm 90^\circ$;
(ii) a hexadecapolar resonance (the "inverse Kozai" resonance) for perturbers
that are circular or nearly so and inclined by $I \simeq 63^\circ/117^\circ$,
in which $\omega_2$ librates about $\pm 90^\circ$ and which can vary the
particle eccentricity by $\Delta e_2 \simeq 0.2$ and lead to orbit crossing;
(iii) an octopole "apse-aligned" resonance at $I \simeq 46^\circ/107^\circ$
wherein $\varpi_2 - \varpi_1$ librates about $0^\circ$ and $\Delta e_2$ grows
with $e_1$; and (iv) an octopole resonance at $I \simeq 73^\circ/134^\circ$
wherein $\varpi_2 + \varpi_1 - 2 \Omega_2$ librates about $0^\circ$ and $\Delta
e_2$ can be as large as 0.3 for small $e_1 \neq 0$. The more eccentric the
perturber, the more the particle's eccentricity and inclination vary; also,
more polar orbits are more chaotic. Our inverse solutions may be applied to the
Kuiper belt and debris disks, circumbinary planets, and stellar systems.
