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

We consider a class of one dimensional Vector Nonlocal Non-linear Schr\"odinger Equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schr\"odinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power, $\cal{PT}$-symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc. are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The cordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.