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We consider a class of one dimensional vector Non-linear Schr$\ddot{o}$dingerEquation(NLSE) in an external complex potential with Balanced Loss-Gain(BLG)and Linear Coupling(LC) among the components of the Schr$\ddot{o}$dinger field.The solvability of the generic system is investigated for various combinationsof time modulated LC and BLG terms, space-time dependent strength of thenonlinear interaction and complex potential. We use a non-unitarytransformation followed by a reformulation of the differential equation in anew coordinate system to map the NLSE to solvable equations. Several physicallymotivated examples of exactly solvable systems are presented for variouscombinations of LC and BLG, external complex potential and nonlinearinteraction. Exact localized nonlinear modes with spatially constant phase maybe obtained for any real potential for which the corresponding linearSchr$\ddot{o}$dinger equation is solvable. A method based on supersymmetricquantum mechanics is devised to construct exact localized nonlinear modes for aclass of complex potentials. The real superpotential corresponding to anyexactly solved linear Schr$\ddot{o}$dinger equation may be used to find acomplex-potential for which exact localized nonlinear modes for the NLSE can beobtained. The solutions with singular phases are obtained for a few complexpotentials.

Non-linear Schr$\ddot{o}$dinger equation with time-dependent balanced loss-gain and space-time modulated non-linear interaction