  About 40 transneptunian binaries (TNBs) have fully determined orbits with
about 10 others being solved except for breaking the mirror ambiguity. Despite
decades of study almost all TNBs have only ever been analyzed with a model that
assumes perfect Keplerian motion (e.g., two point masses). In reality, all TNB
systems are non-Keplerian due to non-spherical shapes, possible presence of
undetected system components, and/or solar perturbations. In this work, we
focus on identifying candidates for detectable non-Keplerian motion based on
sample of 45 well-characterized binaries. We use MultiMoon, a non-Keplerian
Bayesian inference tool, to analyze published relative astrometry allowing for
non-spherical shapes of each TNB system's primary. We first reproduce the
results of previous Keplerian fitting efforts with MultiMoon, which serves as a
comparison for the non-Keplerian fits and confirms that these fits are not
biased by the assumption of a Keplerian orbit. We unambiguously detect
non-Keplerian motion in 8 TNB systems across a range of primary radii, mutual
orbit separations, and system masses. As a proof of concept for non-Keplerian
fitting, we perform detailed fits for (66652) Borasisi-Pabu, possibly revealing
a $J_2 \approx 0.44$, implying Borasisi (and/or Pabu) may be a contact binary
or an unresolved compact binary. However, full confirmation of this result will
require new observations. This work begins the next generation of TNB analyses
that go beyond the point mass assumption to provide unique and valuable
information on the physical properties of TNBs with implications for their
formation and evolution.
