{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}} .
    \end{gathered}$$ We also have the master symmetry (\[eq:phi-sys-msym\]), which can be written $$\begin{gathered}
\partial_\tau \phi^{(0)}_{m,n} = m \partial_{t^1} \phi^{(0)}_{m,n} ,\qquad \partial_\tau \phi^{(1)}_{m,n} = m \partial_{t^1} \phi^{(1)}_{m,n} ,\qquad \partial_\tau \alpha = \alpha,\end{gathered}$$ which allows us to construct a hierarchy of symmetries of system (\[eq:3D-1212\]) in the $m$-direction. For instance, the second symmetry is

\[eq:3D-1212-sym-2\] $$\begin{gathered}
\partial_{t_2} \phi^{(0)}_{m,n} = \frac{\phi^{(0)}_{m,n} \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n}}{{\cal{F}}_{m,n}} ({\cal{S}}_m+1)\left( \frac{({\cal{S}}_m- 1)\big(\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} \phi^{(0)}_{m-2,n} \big)}{{\cal{F}}_{m,n} {\cal{F}}_{m-1,n}} \right),\\
\partial_{t_2} \phi^{(1)}_{m,n} = \frac{\phi^{(1)}_{m,n} \phi^{(0)}_{m+1,n} \phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n}}{{\cal{F}}_{m,n}} ({\cal{S}}_m+1)\left( \frac{({\cal{S}}_m- 1)\big(\phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n} \phi^{(1)}_{m-2,n} \big)}{{\cal{F}}_{m,n} {\cal{F}}_{m-1,n}} \right),\end{gathered}$$ where $$\begin{gathered}
{\cal{F}}_{m,n} := 1+\phi^{(0)}_{m+1,n}\phi^{(0)}_{m,n} \phi^{(0)}_{m-1,n} + \phi^{(1)}_{m+1,n} \phi^{(1)}_{m,n} \phi^{(1)}_{m-1,n} ,\end{gathered}$$ and ${\cal{S}}_m$ denotes the shift operator in the $m$-direction.

### The reduced system {#the-reduced-system .unnumbered}

The reduced system (\[self-dual-equn\]) takes the explicit form (first introduced in [@14-6]) $$\begin{gathered}
 \label{eq:MX2}
\phi_{m,n} \phi_{m+1,n+1} ( \phi_{m+1,n} + \phi_{m,n+1} ) = 2,\end{gathered}$$ where $$\begin{gathered}
 \phi^{(0)}_{m,n} = \phi^{(1)}_{m,n} = \frac{1}{\phi_{m,n}} ,\qquad \beta = - \alpha.\end{gathered}$$ With this coordinate, the second symmetry (\[eq:3D-1212-sym-2\]) takes the form $$\begin{gathered}
\partial_{t_2} \phi_{m,n} = \p