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We consider a class of one dimensional Vector Nonlocal Non-linearSchr\"odinger Equation (VNNLSE) in an external complex potential withtime-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among thecomponents of Schr\"odinger fields, and space-time dependent nonlinearstrength. The system admits Lagrangian and Hamiltonian formulations undercertain conditions. It is shown that various dynamical variables like totalpower, $\cal{PT}$-symmetric Hamiltonian, width of the wave-packet and its speedof growth, etc. are real-valued despite the Hamiltonian density beingcomplex-valued. We study the exact solvability of the generic VNNLSE with orwithout a Hamiltonian formulation. In the first part, we study time-evolutionof moments which are analogous to space-integrals of Stokes variables and findcondition for existence of solutions which are bounded in time. In the secondpart, we use a non-unitary transformation followed by a coordinatetransformation to map the VNNLSE to various solvable equations. The cordinatetransformation is not required at all for the limiting case when non-unitarytransformation reduces to pseudo-unitary transformation. The exact solutionsare bounded in time for the same condition which is obtained through the studyof time-evolution of moments. Various exact solutions of the VNNLSE arepresented.

Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schr\"odinger equation with balanced loss-gain