  Haumea is a dwarf planet with two known satellites, an unusually high spin
rate, and a large collisional family, making it one of the most interesting
objects in the outer solar system. A fully self-consistent formation scenario
responsible for the satellite and family formation is still elusive, but some
processes predict the initial formation of many small moons, similar to the
small moons recently discovered around Pluto. Deep searches for regular
satellites around KBOs are difficult due to observational limitations, but
Haumea is one of the few for which sufficient data exist. We analyze Hubble
Space Telescope (HST) observations, focusing on a ten-consecutive-orbit
sequence obtained in July 2010, to search for new very small satellites. To
maximize the search depth, we implement and validate a non-linear
shift-and-stack method. No additional satellites of Haumea are found, but by
implanting and recovering artificial sources, we characterize our sensitivity.
At distances between $\sim$10,000 km and $\sim$350,000 km from Haumea,
satellites with radii as small as $\sim$10 km are ruled out, assuming an albedo
($p \simeq 0.7$) similar to Haumea. We also rule out satellites larger than
$\gtrsim$40 km in most of the Hill sphere using other HST data. This search
method rules out objects similar in size to the small moons of Pluto. By
developing clear criteria for determining the number of non-linear rates to
use, we find that far fewer shift rates are required ($\sim$35) than might be
expected. The non-linear shift-and-stack method to discover satellites (and
other moving transients) is tractable, particularly in the regime where
non-linear motion begins to manifest itself.
