  We analyze a nonlinear PDE system describing the motion of a microswimmer in
a nematic liquid crystal environment. For the microswimmer's motility, the
squirmer model is used in which self-propulsion enters the model through the
slip velocity on the microswimmer's surface. The liquid crystal is described
using the well-established Beris-Edwards formulation. In previous computational
studies, it was shown that the squirmer, regardless of its initial
configuration, eventually orients itself either parallel or perpendicular to
the preferred orientation dictated by the liquid crystal. Furthermore, the
corresponding solution of the coupled nonlinear system converges to a steady
state. In this work, we rigorously establish the existence of steady state and
also the finite-time existence for the time-dependent problem. Finally, using a
two-scale asymptotic expansion we derive a reduced model for the collective
swimming of squirmers as they reach their steady state orientation and speed.
