Let $U$ and $V$ be non-singular $N \times N$ matrices depending on $(x, t, \lambda)$ which are periodic in $x$ with period $2 \pi$. Consider the associated linear problem

$$\Psi_{x}=U \Psi, \quad \Psi_{t}=V \Psi$$

for the vector $\Psi=\Psi(x, t ; \lambda)$. On the assumption that these equations are compatible, derive the zero curvature equation for $(U, V)$.

Let $W=W(x, t, \lambda)$ denote the $N \times N$ matrix satisfying

$$W_{x}=U W, \quad W(0, t, \lambda)=I_{N}$$

where $I_{N}$ is the $N \times N$ identity matrix. You should assume $W$ is unique. By considering $\left(W_{t}-V W\right)_{x}$, show that the matrix $w(t, \lambda)=W(2 \pi, t, \lambda)$ satisfies the Lax equation

$$w_{t}=[v, w], \quad v(t, \lambda) \equiv V(2 \pi, t, \lambda)$$

Deduce that $\left\{\operatorname{tr}\left(w^{k}\right)\right\}_{k \geqslant 1}$ are first integrals.

By considering the matrices

$$\frac{1}{2 \mathrm{i} \lambda}\left[\begin{array}{cc} \cos u & -\mathrm{i} \sin u \\ \mathrm{i} \sin u & -\cos u \end{array}\right], \quad \frac{\mathrm{i}}{2}\left[\begin{array}{cc} 2 \lambda & u_{x} \\ u_{x} & -2 \lambda \end{array}\right]$$

show that the periodic Sine-Gordon equation $u_{x t}=\sin u$ has infinitely many first integrals. [You need not prove anything about independence.]