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 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 stable. Sufficiently strong Zeeman splitting creates stable semi-vortex states, with vorticities 0 in one component and 2 in the other. mixed modes, which combine fundamental and vortex terms with S 1 = in both field components [7]. Stability of nonlinear states in SOC systems is radically different with respect to the ones without SOC [18,19]. Indeed, the GPE in the free 2D space with the cubic self-attraction, or a system of two GPEs with self-and cross attraction, gives rise to families of zero-vorticity, alias fundamental (Townes' [9]) and vorticity-carrying [10][11][12] bright solitons which are completely unstable either due to the critical collapse [13][14][15] or due to the ringsplitting instabilities [16,17], respectively. On the other hand, the SOC terms come with a coefficient which fixes an inverse length scale in the system, thus breaking the scaling invariance which makes norms of all the solitons belonging to the Townes' family exactly equal to the critical value necessary for the onset of the collapse. As a OPEN ACCESS RECEIVED