vanish, which yields $$K= -\tfrac{1}{2}(u^2-u_x^2)-1, 
\quad
\omega = -\tfrac{1}{2}(u^2-u_x^2)v -u .$$ The matrices – thereby satisfy the zero-curvature equation whenever $v$ is a solution of the mCH equation . In particular, they are given by $$U = \tfrac{1}{2} 
\begin{pmatrix}
1 & v \\ -v & -1
\end{pmatrix}, 
\quad
V = \tfrac{1}{4} 
\begin{pmatrix}
u_x^2-u^2 -2 & (u_x^2-u^2)v -2(u +u_x)
\\
(u^2-u_x^2)v +2(u -u_x) & u^2-u_x^2 +2
\end{pmatrix} .$$ A Lax pair is obtained from these matrices by applying a gauge transformation defined by the mCH scaling symmetry $u\to\lambda^{-1} u$, $x\to x$, $t\to\lambda^{2}t$, combined with the scaling $U\to U$, $V\to\lambda^{-2} V$. This yields $$\begin{aligned}
U & = \tfrac{1}{2} 
\begin{pmatrix}
1 & \lambda v \\ -\lambda v & -1
\end{pmatrix},
\label{mch-U}
\\
V & = \tfrac{1}{4} 
\begin{pmatrix}
u_x^2 - u^2 -2\lambda^{-2} 
& \lambda(u_x^2 -u^2)v -2\lambda^{-1} (u +u_x)
\\
\lambda(u^2-u_x^2)v +2\lambda^{-1} (u -u_x) 
& u^2-u_x^2 + 2\lambda^{-2}
\end{pmatrix},
\label{mch-V}\end{aligned}$$ which is the standard mCH matrix Lax pair (up to rescaling of the spectral parameter $\lambda$).

Thus, we have shown how the AKNS zero-curvature equation can be used to derive the matrix Lax pair for the mCH equation. This derivation has not appeared previously in the literature.

An important final remark arises when we compare 