and an arbitrary function $u(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})
=\tilde{u}(w_1, \cdots, w_{2n+m})$ vanishes such that $$\begin{aligned}
\{G_s, u\}_{\mbox{\tiny{PB}}} 
&=& \frac{1}{2} \sum_{a, b=1}^{2n} 
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, x_b)}\{x_a, x_b\}_{\mbox{\tiny{PB}}}
\nonumber \\
&& 
+ \sum_{a=1}^{2n} \sum_{s=1}^{m} 
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(x_a, z_s)}\{x_a, z_s\}_{\mbox{\tiny{PB}}}
+ \frac{1}{2} \sum_{s, t=1}^{m} 
\frac{\partial(\tilde{G}_s, \tilde{u})}
{\partial(z_s, z_t)}\{z_s, z_t\}_{\mbox{\tiny{PB}}}
\nonumber \\
&=& \sum_{a, b=1}^{2n} 
\tilde{g}^{(m)}_{ab} 
\frac{\partial (\tilde{G}_s, \tilde{u}, \tilde{G}_1, \cdots, \tilde{G}_{m})}
{\partial (x_a, x_b, z_1, \cdots, z_m)} = 0~.
\label{phiu=0}\end{aligned}$$ Hence $G_s(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, 
\cdots, q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ are constants and, if necessary, we can define $G_s=\tilde{G}_s=0$ by shifting constants. We refer to Eq. (\[N-eqf(m)\]) as the generalized Nambu equation. Note that the Liouville theorem does not hold in general for the dynamics given by this equation. This unfavorable property is a result of two factors: Eq. (\[N-eqf(m)\]) has $x_{a}$-dependent $\tilde{g}^{(m)}_{ab}$ and multiplets in Eq. (\[N-eqf(m)\]) share common variables $z_{s}$. The latter means that it is difficult to define an appropriate phase space volume.

One of the non-vanishing components of $\tilde{g}^{(m)}_{ab}$ can be set to $\frac{1}{2}$ by redefinition of constraints $\tilde{G}_s$. For example, in the case in which $n=1$, we can set $\tilde{g}^{(m)}_{12}=\frac{1}{2}$ (and $\tilde{g}^{(m)}_{21}=-\frac{1}{2}$) by redefining $\tilde{G}_s$, and Eq. (\[N-eqf(m)\]) reduces to the Nambu equation (Eq. (\[H-eqf(gN)\])) with $N=2+m$.

