The nonequilibrium and nonlinear defect modes (NNDMs), localized by a defect in a nonequilibrium microcavity-polariton condensate (MPC), are studied. There are three analytic solutions of NNDMs in a point defect: the bright NNDM, a bound state with two dark solitons for an attractive potential, and a gray soliton bound by a defect for a repulsive potential. We find that the stable NNDMs in a nonequilibrium MPC are the bright NNDM and gray soliton bound by a defect. The bright NNDM, which has the hyperbolic cotangent form, is a bright localized state existing in a uniform MPC. The bright NNDM is a unique state occurring in a nonequilibrium MPC that has pump-dissipation and repulsive-nonlinearity characters. No such state can exist in an equilibrium system with repulsive nonlinearity.
