  The discovery of pulsating ultra-luminous X-ray sources (PULX) suggests that
neutron stars are presumably common within the ultra-luminous X-ray source
(ULX) population though the majority of the population members are currently
lacking pulsations. These systems are likely to host neutron stars accreting
mass at super-Eddington (super-critical) rates from their massive companion in
high-mass X-ray binaries. Taking into account the spherization of the accretion
flow in the super-critical regime, the beaming of X-ray emission, and the
reduction of the scattering cross-section in a strong magnetic field, we infer
the ranges for the neutron-star surface magnetic dipole field strengths,
beaming fractions, and fastness parameters in the PULX M82 X-2, ULX NGC 5907,
ULX NGC 7793 P13, NGC 300 ULX1, M51 ULX-7, NGC 1313 X-2, and Swift J0243.6+6124
from a set of conditions based on a variety of combinations of different spin
and luminosity states. Using the observed spin-up rates under the critical
luminosity condition, we estimate the surface-field strengths in the $\sim
10^{11}-10^{13}\,{\rm G}$ range for all PULX. In general, the results of our
analysis under the subcritical luminosity condition indicate surface-field
strengths in the $\sim 10^{11}-10^{15}\,{\rm G}$ range. We argue that the PULX
do not require magnetar-strength surface dipole fields if beaming is taken into
account; yet the fields are strong enough for the neutron stars in ULX to
magnetically channel the accretion flow in super-critical accretion disks.
