hi_{m,n} \frac{1}{P^{(1)}_{m,n}} \left( \frac{1}{P^{(1)}_{m+1,n}}-\frac{1}{P^{(1)}_{m-1,n}}\right),\end{gathered}$$ where $$\begin{gathered}
 \label{eq:N3-P-G}
P^{(0)}_{m,n} = \phi_{m+1,n} \phi_{m,n} \phi_{m-1,n} , \qquad P^{(1)}_{m,n} = 2 + P^{(0)}_{m,n},\end{gathered}$$ first given in [@14-6]. Despite the $t_2$ notation, this is the [*first*]{} of the hierarchy of symmetries of the reduction (\[eq:MX2\]).

The case $\boldsymbol{N=5}$, with level structure $\boldsymbol{(2,3;2,3)}$
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In this case, equations (\[eq:dLP-gen-sys-1-a\]) take the form

\[eq:N5-sys\] $$\begin{gathered}
\phi ^{(0)}_{m+1,n+1}= \frac{\phi ^{(2)}_{m+1,n} \phi ^{(2)}_{m,n+1}}{\phi^{(0)}_{m,n}} \frac{\alpha \phi ^{(3)}_{m+1,n}-\beta \phi ^{(3)}_{m,n+1}}{\alpha \phi ^{(2)}_{m+1,n}- \beta \phi ^{(2)}_{m,n+1}}, \\
\phi ^{(1)}_{m+1,n+1} = \frac{1}{\phi ^{(1)}_{m,n} \big(\alpha \phi ^{(3)}_{m+1,n}-\beta \phi ^{(3)}_{m,n+1}\big)} \left(\frac{\alpha \phi ^{(3)}_{m,n+1}}{\phi ^{(0)}_{m+1,n}\phi ^{(1)}_{m+1,n} \phi^{(2)}_{m+1,n}} -\frac{\beta \phi ^{(3)}_{m+1,n}}{\phi ^{(0)}_{m,n+1} \phi ^{(1)}_{m,n+1} \phi ^{(2)}_{m,n+1}}\right), \nonumber \\
\phi ^{(2)}_{m+1,n+1} = \frac{\alpha \phi ^{(0)}_{m+1,n}-\beta \phi ^{(0)}_{m,n+1}}{\phi ^{(2)}_{m,n}
\big(\alpha \phi ^{(0)}_{m,n+1} \phi ^{(1)}_{m,n+1} \phi ^{(2)}_{m,n+1} \phi ^{(3)}_{m,n+1}-\beta \phi ^{(0)}_{m+1,n} \phi ^{(1)}_{m+1,n} \phi ^{(2)}_{m+1,n} \phi^{(3)}_{m+1,n}\big)}, \!\!\!\\
\phi ^{(3)}_{m+1,n+1} = \frac{\phi ^{(0)}_{m+1,n} \phi ^{(0)}_{m,n+1}}{ \phi ^{(3)}_{m,n}} \frac{\alpha \phi^{(1)}_{m+1,n} - \beta \phi ^{(1)}_{m,n+1}}{\alpha \phi ^{(0)}_{m+1,n}-\beta\phi ^{(0)}_{m,n+1}}.\end{gathered}$$

Under the transformation (\[sd\]), the first and last of these interchange, as do the middle pair.

The lowest order generalised symmetry in the $m$-direction is generated by $$\begin{gathered}
\partial_{t_1} \phi^{(i)}_{m,n} = \phi^{(i)}_{m,n} \left(\frac{5 A^{(i)}_{m,n}}{B_{m,n}}-1\right),\qquad i=0,\ldots,3,\end{gathered}$$ whe