  We investigate the properties of a two-dimensional \emph{spinor} microcavity
polariton system driven by a linearly polarised continuous pump. In particular,
we establish the role of the elementary excitations, namely the so-called
half-vortices and full-vortices; these objects carry a quantum rotation only in
one of the two, or both, spin components respectively. Our numerical analysis
of the steady-state shows that it is only the half-vortices that are present in
the vortex-antivortex pairing/dissociation responsible for the
Berezinskii-Kosterlitz-Thouless transition. These are the relevant elementary
excitations close to the critical point. However, by exploring the
phase-ordering dynamics following a sudden quench across the transition we
prove that full-vortices become the relevant excitations away from the critical
point in a deep quasi-ordered state at late times. The time-scales for
half-vortices binding into full vortices are much faster than the
vortex-antivortex annihilations.
