  We consider the three currently known pulsing ultraluminous X--ray sources
(PULXs). We show that in one of them the observed spinup rate requires
super--Eddington accretion rates at the magnetospheric radius, even if
magnetar--strength fields are assumed. In the two other systems a
normal--strength neutron star field implies super--Eddington accretion at the
magnetosphere. Adopting super--Eddington mass transfer as the defining
characteristic of ULX systems, we find the parameters required for
self--consistent simultaneous fits of the luminosities and spinup rates of the
three pulsed systems. These imply near--equality between their magnetospheric
radii $R_M$ and the spherization radii $R_{\rm sph}$ where radiation pressure
becomes important and drives mass loss from the accretion disc. We interpret
this near--equality as a necessary condition for the systems to appear as
pulsed, since if it is violated the pulse fraction is small. We show that as a
consequence all PULXs must have spinup rates $\dot\nu \gtrsim 10^{-10}\, {\rm
s^{-2}}$, an order of magnitude higher than in any other pulsing neutron--star
binaries. The fairly tight conditions required for ULXs to show pulsing support
our earlier suggestion that many unpulsed ULX systems must actually contain
neutron stars rather than black holes.
