${\cal CPT}$ -symmetric spin-orbit–coupled condensate

We consider a mean-field model of a spin-orbit–coupled atomic system (Bose-Einstein condensate) accounting for pumping and removal of atoms from the two ground states. This model introduces a concept of charge-parity-time -symmetric system with a complex potential featuring a purely real spectrum. The introduced spin-orbit–coupled BEC possesses properties such as control of the stability by the external trap, singular symmetry phase breaking, the existence of stable nonlinear modes with nonzero currents in repulsive and attractive condensates, multiply re-entrant -symmetric phase, splitting of nonlinear modes into sets of solitons (wave packets) moving with different velocities and in different directions in the condensate released from the trap.