phi^{(0)}_{m,n}\big)^2 \phi^{(0)}_{m+1,n} \phi^{(1)}_{m-1,n} \big(\phi^{(1)}_{m,n}\big)^2 \phi^{(1)}_{m+1,n} \phi^{(2)}_{m-1,n} \phi^{(2)}_{m,n} \phi^{(2)}_{m+1,n},\\
    P^{(3)}_{m,n} = 2 + 2 P^{(0)}_{m,n} + 2 P^{(1)}_{m,n}+ P^{(2)}_{m,n}.
    \end{gathered}$$

Miura transformations and relation to Bogoyavlensky lattices {#sec:Miura}
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In this section we discuss Miura transformations for the reduced systems and their symmetries which bring the latter to polynomial form. In the lowest dimensional case ($N=3$) the polynomial system is directly related to the Bogoyavlensky lattice (see (\[eq:Bog\]) below), whereas the higher dimensional ones result in systems which generalise (\[eq:Bog\]) to $k$ component systems.

The reduced system in $\boldsymbol{N=3}$ {#the-reduced-system-in-boldsymboln3 .unnumbered}
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The Miura transformation [@14-6] $$\begin{gathered}
 \psi_{m,n} = \frac{P^{(0)}_{m,n}}{P^{(1)}_{m,n}} - 1,\end{gathered}$$ where $P^{(0)}_{m,n}$ and $P^{(1)}_{m,n}$ are given in (\[eq:N3-P-G\]), maps equation (\[eq:MX2\]) to $$\begin{gathered}
 \label{eq:MX2a}
\frac{\psi_{m+1,n+1}+1}{\psi_{m,n}+\psi_{m,n+1}+1} + \frac{\psi_{m+1,n}}{\psi_{m,n+1}} = 0,\end{gathered}$$ and its symmetry to $$\begin{gathered}
\label{eq:Bog}
\partial_{t_2} \psi_{m,n} = \psi_{m,n} (\psi_{m,n}+1) (\psi_{m+2,n} \psi_{m+1,n} - \psi_{m-1,n} \psi_{m-2,n}),\end{gathered}$$ which is related to the Bogoyavlensky lattice [@B] $$\begin{gathered}
 \partial_{t_2} \chi_{m,n} = \chi_{m,n}(\chi_{m+2,n} + \chi_{m+1,n} - \chi_{m-1,n} - \chi_{m-2,n}),\end{gathered}$$ through the Miura transformation $$\begin{gathered}
 \chi_{m,n} = \psi_{m+1,n} \psi_{m,n} (\psi_{m-1,n}+1).\end{gathered}$$

The reduced system in $\boldsymbol{N=5}$ {#the-reduced-system-in-boldsymboln5 .unnumbered}
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The Miura transformation $$\begin{gathered}
\psi_{m,n}^{(0)} = \frac{2 P^{(0)}_{m,n}}{P^{(2)}_{m,n}},\qquad \psi_{m,n}^{(1)} = \frac{P^{