Let $U(\rho, \tau, \lambda)$ and $V(\rho, \tau, \lambda)$ be matrix-valued functions. Consider the following system of overdetermined linear partial differential equations:

$$\frac{\partial}{\partial \rho} \psi=U \psi, \quad \frac{\partial}{\partial \tau} \psi=V \psi$$

where $\psi$ is a column vector whose components depend on $(\rho, \tau, \lambda)$. Using the consistency condition of this system, derive the associated zero curvature representation (ZCR)

$$\frac{\partial}{\partial \tau} U-\frac{\partial}{\partial \rho} V+[U, V]=0,$$

where $[\cdot, \cdot]$ denotes the usual matrix commutator.

(i) Let

$$U=\frac{i}{2}\left(\begin{array}{cc} 2 \lambda & \partial_{\rho} \phi \\ \partial_{\rho} \phi & -2 \lambda \end{array}\right), \quad V=\frac{1}{4 i \lambda}\left(\begin{array}{cc} \cos \phi & -i \sin \phi \\ i \sin \phi & -\cos \phi \end{array}\right)$$

Find a partial differential equation for $\phi=\phi(\rho, \tau)$ which is equivalent to the $\mathrm{ZCR}(*)$.

(ii) Assuming that $U$ and $V$ in $(*)$ do not depend on $t:=\rho-\tau$, show that the trace of $(U-V)^{p}$ does not depend on $x:=\rho+\tau$, where $p$ is any positive integer. Use this fact to construct a first integral of the ordinary differential equation

$$\phi^{\prime \prime}=\sin \phi, \quad \text { where } \quad \phi=\phi(x) .$$