of the Hamiltonian system $(q_{\mbox{\tiny(1)}}, p_{\mbox{\tiny(1)}}, \cdots, 
q_{\mbox{\tiny({\it n})}}, p_{\mbox{\tiny({\it n})}})$ is defined as $$\begin{aligned}
Z_{\rm H} \equiv 
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int 
\prod_{{\rm k}=1}^{n} dq_{\mbox{\tiny(k)}} dp_{\mbox{\tiny(k)}} e^{-\beta H}~,
\label{ZH}\end{aligned}$$ where $\beta = 1/(k_BT)$ is the inverse temperature made up of the Boltzmann constant $k_B$ and the temperature $T$. Here we study the partition function $Z_{\rm N}$ for an ensemble of the Nambu system $(x_{\mbox{\tiny(1)}}, y_{\mbox{\tiny(1)}}, \cdots, z_{\mbox{\tiny({\it n})}})$ hidden in the Hamiltonian system.

First let us conjecture the form of $Z_{\rm N}$ on physical grounds. Since $\tilde{H}=H$, $Z_{\rm N}$ must contain the $\lq\lq$Boltzmann weight" such as $e^{-\beta \tilde{H}}$. The other Hamiltonian $\tilde{G}$ is the sum of the constraints $\tilde{G}_{\mbox{\tiny(k)}} 
(x_{\mbox{\tiny(k)}}, y_{\mbox{\tiny(k)}}, z_{\mbox{\tiny(k)}})
= G_{\mbox{\tiny(k)}} (q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})= 0$, and therefore there should be delta functions such as $\delta(\tilde{G}_{\mbox{\tiny(k)}})$ in $Z_{\rm N}$. Furthermore, $Z_{\rm N}$ must contain the volume element $\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}$ from the Liouville theorem.

On the basis of the above observations, it is expected that $Z_{\rm N}$ should have a form such that $$\begin{aligned}
Z_{\rm N} &\equiv& 
\iint\!\!\cdot\!\cdot\!\cdot\!\!\int 
\prod_{{\rm k}=1}^{n}
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}} 
\delta(\tilde{G}_{\mbox{\tiny(k)}}) e^{-\beta \tilde{H}}
\label{ZN1}\\
&=& \iint\!\!\cdot\!\cdot\!\cdot\!\!\int  
\prod_{{\rm k}=1}^{n} 
dx_{\mbox{\tiny(k)}} dy_{\mbox{\tiny(k)}} dz_{\mbox{\tiny(k)}}  
\int_{-\infty}^{\infty} \frac{d\gamma_{\mbox{\tiny(k)}}}{2\pi} 
e^{-\beta \tilde{H}
-i \sum_{\rm k} \gamma_{\mbox{\tiny(k)}} \tilde{G}_{\mbox{\tiny(k)}}}~.
\label{ZN2}\end{aligned}$$ We can derive $Z_{\rm H}$ (Eq. 