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The linearized perturbation obeys $\dot{\eta }=D\mathbf{f}(\mathbf{X}(t))\eta$, so its fundamental matrix satisfiesThe Floquet multipliers are the eigenvalues of the monodromy matrix $\Phi (T)$. Differentiating the orbit equation shows that $\dot{\mathbf{X}}(t)=\mathbf{f}(\mathbf{X}(t))$ is a solution of the variational equation. For a nonconstant periodic orbit this tangent solution is nonzero and periodic, giving one multiplier equal to $1$.

The Liouville formula for a fundamental matrix gives $(\det \Phi {)}^{\prime }=(\mathrm{tr}D\mathbf{f})\det \Phi$. Since $\det \Phi (0)=1$, the product of the two multipliers is $\exp ({\int }_0^T\nabla \cdot \mathbf{f}(\mathbf{X}(t))dt)$. The other multiplier is therefore exactly this exponential.

Compatibility of the auxiliary system, computed on a fundamental matrix of solutions, gives the zero-curvature conditionIndeed equality of the mixed derivatives gives $(U_t+UV-V_x-VU)\Psi =0$. A single possibly zero vector solution would not justify the matrix identity; compatibility means a full fundamental family.

Put $H=W_t-VW$. Using the zero-curvature condition gives $H_x=UH$. At $x=0$, $W_t=0$ and hence $H(0)=-V(0)$. Uniqueness of the initial-value problem gives $H=-WV(0)$, or $W_t=VW-WV(0)$. Periodicity implies $V(2\pi )=V(0)=v$, so the monodromy matrix $w=W(2\pi )$ obeys the Lax equationCyclicity of the trace yields $\frac{d}{dt}\mathrm{tr}(w^k)=k\mathrm{tr}(w^{k-1}[v,w])=0$ for every $k\ge 1$.

There is a genuine factor-of-two error in the printed sine-Gordon pair. Assign the second printed matrix to $U$ and the first to $V_{\mathrm{print}}$:Direct multiplication givesThus the printed matrices encode $u_{xt}=2\sin u$.

For the stated Sine-Gordon equation, explicitly repair the first prefactor by taking $V=V_{\mathrm{print}}/2$, that is $1/(4i\lambda )$. Its zero-curvature condition becomes $u_{xt}=\sin u$. For this corrected pair, $\mathrm{tr}(w(t,\lambda {)}^k)$ is conserved for every $\lambda \ne 0$. Since $U$ is entire in $\lambda$, so are $W$ and these traces. Their Taylor coefficients in $\lambda$ supply an infinite family of first integrals, by the identity theorem extending conservation through $\lambda =0$. At a fixed spectral parameter the higher traces of a $2\times 2$ matrix are related by the Cayley-Hamilton theorem; no independence of the resulting first integrals is claimed.