  Aims: Main goal of this work is to study the potential of He II Ly-alpha
wavelength-integrated scattering polarisation for probing the magnetism of the
solar upper chromosphere.
  Methods: Radiative transfer calculations are performed in semi-empirical 1D
solar atmospheres considering a two-term atomic model and accounting for the
Hanle, Zeeman, and magneto-optical effects. The problem is suitably linearised
and discretised, and the resulting numerical system is solved with a
matrix-free iterative method. The results obtained modelling scattering
processes with three different descriptions, namely in the limit of complete
frequency redistribution (CRD), and accounting for partial frequency
redistribution (PRD) effects under the angle-averaged (AA) approximation and in
the general angle-dependent (AD) formulation, are compared.
  Results: In the line-core, the synthetic Stokes profiles resulting from CRD,
PRD-AA, and PRD-AD calculations show a very good agreement. On the other hand,
relevant differences are observed in Q/I outside the line-core region. Besides,
the precise structure of the atmospheric model does not noticeably affect the
line-core profiles, but it strongly impacts the Q/I signals outside the
line-core. As most of the He II Ly-alpha photons originate in the core region,
it turns out that wavelength-integrated linear polarisation signals are almost
insensitive to both the scattering description and the atmospheric model.
Appreciable wavelength-integrated U/I signals, showing observable sensitivity
to horizontal magnetic fields in the range 0-1000 G, are also found,
particularly near the limb. It turns out that, while the integration time
required to detect magnetic fields in the quiet chromosphere with this line is
too long for sounding rocket missions, magnetic fields corresponding to typical
plage areas would produce detectable signals, especially near the limb.
