  LS 5039 is one of a handful of X-ray binaries that have been recently
detected at high-energy $\gamma$-rays, in this case, by the High-Energy
Stereoscopy Array (H.E.S.S.). The nature of this system is unknown: both a
black hole and a pulsar have been invoked as possible compact object
companions. Here we work with a model of the high energy phenomenology of the
system in which it is assumed that the companion object is a pulsar rotating
around an O6.5V star in the $\sim 3.9$ days orbit. The model assumes two
different sets of power-law spectral parameters of the interacting primary
leptons corresponding to the two orbital phase intervals defined by H.E.S.S. as
having different gamma-ray spectra and very-high-energy (VHE) cutoffs. We show
the H.E.S.S. phenomenology is completely explained by this model. We present
predictions for photons with lower energies (for $E>1 $ GeV), subject to test
in the forthcoming months with the GLAST satellite. We find that GLAST is able
to judge on this model within one year.
