  Discovery of pulsations from a number of ultra-luminous X-ray (ULX) sources
proved that accretion onto neutron stars can produce luminosities exceeding the
Eddington limit by a couple of orders of magnitude. The conditions necessary to
achieve such high luminosities as well as the exact geometry of the accretion
flow in the neutron star vicinity are, however, a matter of debate. The pulse
phase-resolved polarization measurements that became possible with the launch
of the IXPE can be used to determine the pulsar geometry and its orientation
relative to the orbital plane. They provide an avenue to test different
theoretical models of ULX pulsars. In this paper we present the results of
three IXPE observations of the first Galactic ULX pulsar Swift J0243.6+6124
during its 2023 outburst. We find strong variations of the polarization
characteristics with the pulsar phase. The average polarization degree
increases from about 5% to 15% as the flux dropped by a factor of three in the
course of the outburst. The polarization angle (PA) as function of the pulsar
phase shows two peaks in the first two observations, but changes to a
characteristic sawtooth pattern in the remaining data set. This is not
consistent with a simple rotating vector model. Assuming the existence of an
additional constant polarized component, we were able to fit the three
observations with a common rotating vector model and obtain constraints on the
pulsar geometry. In particular, we find the pulsar angular momentum inclination
with respect to the line-of-sight of 15-40 deg, the magnetic obliquity of 60-80
deg, and the pulsar spin position angle of -50 deg, which differs from the
constant component PA of about 10 deg. Combining these X-ray measurements with
the optical PA, we find evidence for a 30 deg misalignment between the pulsar
spin and the binary orbital axis.
