  When studying the dynamics of Vertical-Cavity Surface-Emitting Lasers, and
their polarization properties, the spin-flip model appears to be the simplest
model qualitatively reproducing all dynamical features that have been observed
experimentally. Nonetheless, because of the fast time-scale of the
spin-relaxation processes, the specific role and the importance of the
spin-population difference - which is one of the specific feature of the
spin-flip model - has been continuously questioned. In fact, the debate
regarding the possible adiabatic elimination of the spin-population remains
fully open.
  In this paper, our goal is to bring new light into this issue by
demonstrating that this variable is essential to preserve the most complex
dynamical features predicted by the spin-flip model, such as polarization
chaos, and, therefore, needs to be conserved. To do so, we first perform a
detailed analysis, focusing on the chaotic dynamics, to determine the minimal
embedding dimension for the spin-flip model. As the latter confirms that a
reduction of the model could be envisaged, we then consider the adiabatic
elimination of the spin-population difference to highlight and explain its
failure to reproduce essential dynamical features obtained in the original
model.
