We consider a class of one dimensional vector Non-linear Schr$\ddot{o}$dinger Equation(NLSE) in an external complex potential with Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components field. The solvability generic system is investigated for various combinations time modulated LC BLG terms, space-time dependent strength nonlinear interaction potential. use non-unitary transforma...
