$\R-1$. This operator arises in the following way from the AKNS scheme [@AblKauNewSeg] using a matrix zero-curvature equation $$\label{zero-curv}
U_t-V_x +[U,V]=0$$ given by the matrices $$\begin{aligned}
U & = 
\begin{pmatrix}
\lambda & \tfrac{1}{2} v
\\
-\tfrac{1}{2} v & -\lambda
\end{pmatrix}, 
\label{real-akns-U}
\\
V & = 
\begin{pmatrix}
\lambda K & \tfrac{1}{2} \omega + \lambda P
\\
-\tfrac{1}{2} \omega + \lambda P & -\lambda K 
\end{pmatrix}
\label{real-akns-V}\end{aligned}$$ where $\lambda$ is the spectral parameter. These matrices belong to the Lie algebra $sl(2,\Rnum)$, with $U$ being a reduction of the standard AKNS matrix to the case of a real spectral parameter $\lambda$ and a real variable $v$, and with $V$ being parameterized such that $\begin{pmatrix} K & 0 \\ 0 & -K \end{pmatrix}$, $\begin{pmatrix} 0 & \omega \\ -\omega & 0 \end{pmatrix}$, $\begin{pmatrix} 0 & P \\ P & 0 \end{pmatrix}$ are mutually orthogonal in the Cartan-Killing inner product (as defined by the trace of a matrix product) in $sl(2,\Rnum)$. The components of the zero-curvature equation yield

$$\begin{gathered}
v_t = D_x \omega -4\lambda^2 P , 
\label{real-akns-flow-eqn}
\\
\omega = v K + D_x P , 
\label{real-akns-grad-eqn}
\\
D_x K = v P . 
\label{real-akns-aux-eqn}\end{gathered}$$

The second and third equations can be used to express 