  We present a model for a super-Eddington accretion disc around a magnetized
neutron star taking into account advection of heat and the mass loss by the
wind. The model is semi-analytical and predicts radial profiles of all basic
physical characteristics of the accretion disc. The magnetospheric radius is
found as an eigenvalue of the problem. When the inner disc is in
radiation-pressure-dominated regime but does not reach its local Eddington
limit, advection is mild, and the radius of the magnetosphere depends weakly on
the accretion rate. Once approaching the local Eddington limit, the disc
becomes advection-dominated, and the scaling for the magnetospheric radius with
the mass accretion rate is similar to the classical Alfven relation. Allowing
for the mass loss in a wind leads to an increase of the magnetospheric radius.
Our model may be applied to a large variety of magnetized neutron stars
accreting close to or above their Eddington limits: ultra-luminous X-ray
pulsars, Be/X-ray binaries in outbursts, and other systems. In the context of
our model we discuss the observational properties of NGC 5907~X-1, the
brightest ultra-luminous pulsar known so far, and NGC 300~ULX-1 which is
apparently a Be/X-ray binary experiencing a very bright super-Eddington
outburst.
