  The dwarf planet Haumea is one of the most compelling transneptunian objects
(TNOs) to study, hosting two small, dynamically interacting satellites, a
family of nearby spectrally unique objects, and a ring system. Haumea itself is
extremely oblate due to its 3.9 hour rotation period. Understanding the orbits
of Haumea's satellites, named Hi'iaka and Namaka, requires detailed modeling of
both satellite-satellite gravitational interactions and satellite interactions
with Haumea's nonspherical gravitational field (parameterized here as $J_2$).
Understanding both of these effects allows for a detailed probe of the
satellites' masses and Haumea's $J_2$ and spin pole. Measuring Haumea's $J_2$
provides information about Haumea's interior, possibly determining the extent
of past differentation. In an effort to understand the Haumea system, we have
performed detailed non-Keplerian orbit fitting of Haumea's satellites using a
decade of new ultra-precise observations. Our fits detect Haumea's $J_2$ and
spin pole at $\gtrsim2.5\sigma$ confidence. Degeneracies present in the
dynamics prevent us from precisely measuring Haumea's $J_2$ with the current
data, but future observations should enable a precise measurement. Our
dynamically determined spin pole shows excellent agreement with past results,
illustrating the strength of non-Keplerian orbit fitting. We also explore the
spin-orbit dynamics of Haumea and its satellites, showing that axial precession
of Hi'iaka may be detectable over decadal timescales. Finally, we present an
ephemeris of the Haumea system over the coming decade, enabling high-quality
observations of Haumea and its satellites for years to come.
