Finally, we consider the case in which the variables $x_a$ and $z_s$ are further classified into $M$ $\lq\lq$irreducible“ sets, $\{x_{a^1}^{\mbox{\tiny{({\rm 1})}}}, 
z_{s^1}^{\mbox{\tiny{({\rm 1})}}}\} 
\bigoplus 
\{x_{a^2}^{\mbox{\tiny{({\rm 2})}}}, 
z_{s^2}^{\mbox{\tiny{({\rm 2})}}}\} 
\bigoplus \cdots \bigoplus 
\{x_{a^M}^{\mbox{\tiny{({\it M})}}}, 
z_{s^M}^{\mbox{\tiny{({\it M})}}}\}$, where $a^i=1,\cdots,2n^i$ ($\sum_{i=1}^{M}n^i=n$) and $s^i=1,\cdots,m^i$ ($\sum_{i=1}^{M}m^i=m$). Here $\lq\lq$irreducible” means that the Poisson bracket between any two elements that belong to different sets vanishes, i.e., $\{x_{a^i}^{\mbox{\tiny{({\it i})}}}, 
x_{a^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, $\{x_{a^i}^{\mbox{\tiny{({\it i})}}}, 
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$, and $\{z_{s^i}^{\mbox{\tiny{({\it i})}}}, 
z_{s^j}^{\mbox{\tiny{({\it j})}}}\}_{\mbox{\tiny{PB}}}=0$ for $i \ne j$. Note that this classification is not unique, either. The equation of motion for any function $\tilde{f}(w_1, \cdots, w_{2n+m})$ can be expressed in the form of the generalized Nambu equation, $$\begin{aligned}
\frac{d \tilde{f}}{dt}
= \sum_{i=1}^{M}
\sum_{a^i, b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^i b^i} 
\frac{\partial (\tilde{f}, \tilde{H}, 
\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}}, 
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (x_{a^i}^{\mbox{\tiny{({\it i})}}}, 
x_{b^i}^{\mbox{\tiny{({\it i})}}}, 
z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots, 
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~.
\label{N-eqf(m)2}\end{aligned}$$ Here $\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}}$ and $\tilde{g}^{(m^i)}_{a^ib^i}$ should satisfy the following conditions: $$\begin{aligned}
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}}, 
x_{b^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=& 
\tilde{g}^{(m^i)}_{a^ib^i} 
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}}, 
\cdots, \tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots, 
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m4}\\
\frac{1}{2} \{x_{a^i}^{\mbox{\tiny{({\it i})}}}, 
z_{s^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}
&=& 
-\sum_{b^i=1}^{2n^i} \tilde{g}^{(m^i)}_{a^ib^i} 
\frac{\partial (\tilde{G}_{1}^{\mbox{\tiny{({\it i})}}}, 
\cdots, \tilde{G}_{s^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{s^i}^{\mbox{\tiny{({\it i})}}}, 
\tilde{G}_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots, 
\tilde{G}_{m^i}^{\mbox{\tiny{({\it i})}}})}
{\partial (z_{1}^{\mbox{\tiny{({\it i})}}}, \cdots, 
z_{s^i-1}^{\mbox{\tiny{({\it i})}}}, 
x_{b^i}^{\mbox{\tiny{({\it i})}}}, 
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m5}\\
\{z_{s^i}^{\mbox{\tiny{({\it 