(a) Let $U(z, \bar{z}, \lambda)$ and $V(z, \bar{z}, \lambda)$ be matrix-valued functions, whilst $\psi(z, \bar{z}, \lambda)$ is a vector-valued function. Show that the linear system

$$\partial_{z} \psi=U \psi, \quad \partial_{\bar{z}} \psi=V \psi$$

is over-determined and derive a consistency condition on $U, V$ that is necessary for there to be non-trivial solutions.

(b) Suppose that

$$U=\frac{1}{2 \lambda}\left(\begin{array}{cc} \lambda \partial_{z} u & e^{-u} \\ e^{u} & -\lambda \partial_{z} u \end{array}\right) \quad \text { and } \quad V=\frac{1}{2}\left(\begin{array}{cc} -\partial_{\bar{z}} u & \lambda e^{u} \\ \lambda e^{-u} & \partial_{\bar{z}} u \end{array}\right)$$

where $u(z, \bar{z})$ is a scalar function. Obtain a partial differential equation for $u$ that is equivalent to your consistency condition from part (a).

(c) Now let $z=x+i y$ and suppose $u$ is independent of $y$. Show that the trace of $(U-V)^{n}$ is constant for all positive integers $n$. Hence, or otherwise, construct a non-trivial first integral of the equation

$$\frac{d^{2} \phi}{d x^{2}}=4 \sinh \phi, \quad \text { where } \quad \phi=\phi(x)$$