We consider vector Non-linear Schrödinger Equation (NLSE) with balanced loss-gain (BLG), linear coupling (LC) and a general form of cubic nonlinearity. We use a non-unitary transformation to show that the system can be exactly mapped to the same equation without the BLG and LC, and with a modified time-modulated nonlinear interaction. The nonlinear term remains invariant, while BLG and LC are removed completely, for the special case of a pseudo-unitary transformation. The mapping is generic and may be used to construct exactly solvable autonomous as well as non-autonomous vector NLSE with BLG. We present an exactly solvable two-component vector NLSE with BLG which exhibits power-oscillation. An example of a vector NLSE with BLG and arbitrary even number of components is also presented.

我们考虑具有平衡损耗增益（BLG），线性耦合（LC）和三次非线性的一般形式的矢量非线性Schrödinger方程（NLSE）。我们使用非-变换表明该系统可以精确映射到相同的方程，而无需BLG和LC，并且具有经过修改的时间调制非线性相互作用。对于伪-变换的特殊情况，非线性项保持不变，而BLG和LC则被完全删除。映射是通用的，可用于构造具有BLG的可精确求解的自治矢量和非自治矢量NLSE。我们提出了一个具有BLG的可精确求解的两分量矢量NLSE，它展现了功率振荡。还给出了具有BLG和任意偶数分量的矢量NLSE的示例。