  We consider a model of the exciton-polariton condensate based on a system of
two Gross-Pitaevskii equations coupled by the second-order differential
operator, which represents the spin-orbit coupling (SOC) in the system. Also
included are the linear gain, effective diffusion, nonlinear loss, and the
standard harmonic-oscillator trapping potential, as well as the Zeeman
splitting. By means of combined analytical and numerical methods, we identify
stable two-dimensional modes supported by the nonlinear system. In the absence
of the Zeeman splitting, these are mixed modes, which combine zero and nonzero
vorticities in each of the two spinor components, and vortex-antivortex
complexes. We have also found a range of parameters where the mixed-mode and
vortex-antivortex states coexist and are simultaneously stable. Sufficiently
strong Zeeman splitting creates stable semi-vortex states, with vorticities 0
in one component and 2 in the other.
