Let $U=U(x, y)$ and $V=V(x, y)$ be two $n \times n$ complex-valued matrix functions, smoothly differentiable in their variables. We wish to explore the solution of the overdetermined linear system

$$\frac{\partial \mathbf{v}}{\partial y}=U(x, y) \mathbf{v}, \quad \frac{\partial \mathbf{v}}{\partial x}=V(x, y) \mathbf{v}$$

for some twice smoothly differentiable vector function $\mathbf{v}(x, y)$.

Prove that, if the overdetermined system holds, then the functions $U$ and $V$ obey the zero curvature representation

$$\frac{\partial U}{\partial x}-\frac{\partial V}{\partial y}+U V-V U=0$$

Let $u=u(x, y)$ and

$$U=\left[\begin{array}{cc} i \lambda & i \bar{u} \\ i u & -i \lambda \end{array}\right], \quad V=\left[\begin{array}{cc} 2 i \lambda^{2}-i|u|^{2} & 2 i \lambda \bar{u}+\bar{u}_{y} \\ 2 i \lambda u-u_{y} & -2 i \lambda^{2}+i|u|^{2} \end{array}\right]$$

where subscripts denote derivatives, $\bar{u}$ is the complex conjugate of $u$ and $\lambda$ is a constant. Find the compatibility condition on the function $u$ so that $U$ and $V$ obey the zero curvature representation.