I only need help with the last part of the following question:
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 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 $$ \phi^{\prime \prime}=\sin \phi, \quad \text { where } \quad \phi=\phi(x) . $$
In particular, in part (ii) I do not see how to show that the trace is zero. I got to show that $$ \frac{d}{dx} (U-V)^p=p[U,V] (U-V)^{p-1}. $$ However, I am not sure if the trace of this is necessarily zero.
For the last part, I assume that I am also meant to combine the trace fact and (i) to get a first integral. But how does zero curvature give us a first integral?
Question How do I show that the trace is invariant of $x$, and moreover, how does one use it to find first integrals?