Integrable local and non-local vector Non-linear Schrödinger Equation with balanced loss and gain

The local and non-local vector Non-linear Schrodinger Equation (NLSE) with a general cubic non-linearity are considered in presence of linear term characterized, general, by non-hermitian matrix which under certain condition incorporates balanced loss gain coupling between the complex fields governing non-linear equations. It is shown that systems posses Lax pair an infinite number conserved quantities hence integrable. Apart from particular form reductions, integrable when representing pseudo hermitian respect to comprising generic non-linearity. inverse scattering transformation method employed find exact soliton solutions for both cases. restricts possible norming constants polarization vector. NLSE term, characterized pseudo-hermitian matrix, selects class corresponding without map it solution via unitary transformation,