[@N]. This comes from the fact that the Nambu mechanics considered here is an effective one induced by the redundancy of the variables.

Finally, we just give the result for the case of general $N(\ge 3)$. The possible form of the partition function is given by $$\begin{aligned}
Z_{\rm N} = \iint\!\!\cdot\!\cdot\!\cdot\!\!\int
\prod_{{\rm k}=1}^{n} 
dx_{1\mbox{\tiny{(k)}}} dx_{2\mbox{\tiny{(k)}}} 
\cdots dx_{N\mbox{\tiny{(k)}}} 
\delta(\tilde{G}_{1\mbox{\tiny{(k)}}}) \delta(\tilde{G}_{2\mbox{\tiny{(k)}}}) 
\cdots \delta(\tilde{G}_{N-2\mbox{\tiny{(k)}}}) e^{-\beta \tilde{H}}~,
\label{ZN-N}\end{aligned}$$ where $\tilde{G}_{b\mbox{\tiny{(k)}}}=0$ $(b = 1, 2, \cdots, N-2)$ are induced constraints. This expression should agree with $Z_{\rm H}$ (Eq. (\[ZH\])) up to some constant normalization factor.

Generalized Nambu equations {#Generalized Nambu equations}
---------------------------

We generalize our formulation to include a specific case that all multiplets share some variables. In such a case, a generalization of the Nambu equation would be required.

Let us describe a Hamiltonian system with $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $({\rm k}=1, \cdots, n)$ using $2n+m$ variables $w_{\ell}$ $(\ell=1, \cdots, 2n+m)$. We classify the variables $w_{\ell}$ into two groups, $x_{a}$ $(a=1, \cdots, 2n)$ and $z_s$ $(s=1, \cdots, m)$, where $x_a$ are assumed to satisfy $\det \{x_a, x_b\}_{\mbox{\tiny{PB}}} \ne 0$. Note that the classification of variables is not unique.

First we study the case with $m=0$ for completeness. In this case, the 