\\
&&\{x_3, x_4\}_{\mbox{\tiny{PB}}} = 
\frac{\partial x_3}{\partial q} \frac{\partial x_4}{\partial p}
- \frac{\partial x_3}{\partial p} \frac{\partial x_4}{\partial q}~.
\label{Ex2-PB}\end{aligned}$$ Then the conditions (Eq. (\[xiGb\])) become $$\begin{aligned}
\sum_{i_3, i_4=1}^{4} \varepsilon_{i_1 i_2 i_3 i_4}
\frac{\partial \tilde{G}_1}{\partial x_{i_3}} 
\frac{\partial \tilde{G}_2}{\partial x_{i_4}}
= \{x_{i_1}, x_{i_2}\}_{\mbox{\tiny{PB}}}~,
\label{Ex2-dG}\end{aligned}$$ and $\tilde{G}_1$ and $\tilde{G}_2$ are given by $$\begin{aligned}
\tilde{G}_1= x_3-x_3(x_1, x_2)+C_1~,~~ \tilde{G}_2= x_4-x_4(x_1, x_2)+C_2~.
\label{Ex2-G}\end{aligned}$$ where $C_1$ and $C_2$ are constants. By redefining $G_1$ and $G_2$ to eliminate the constants, we obtain the induced constraints $\tilde{G}_1(x_1,x_2,x_3,x_4)=G_1(q,p)=0$ and $\tilde{G}_2(x_1,x_2,x_3,x_4)=G_2(q,p)=0$.

Many degrees of freedom systems {#Many degrees of freedom systems}
-------------------------------

Let us extend our formulation to Hamiltonian systems with many degrees of freedom. Consider a Hamiltonian system described by $n$ sets of canonical doublets $(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ (k$=1, 2, \cdots, n$). As is the case with $n=1$ given in Sect. 2.2, hidden Nambu structure can also be found in this system. Here we present the $N=3$ case, i.e., the case with $n$ sets of triplets $x_{i{\mbox{\tiny(k)}}} = x_{i{\mbox{\tiny(k)}}}
(q_{\mbox{\tiny(k)}}, p_{\mbox{\tiny(k)}})$ $(i=1,2,3)$. Generalization to the $N(\ge 3)$ cases is straightforward.

### Dynamics {#Dynamics}

In this system, the Poisson bracket of $A$ and $B$ is defined as $$\begin{aligned}
\{A, B\}_{\mbox{\tiny{PB}}} 
\equiv \sum_{{\rm k}=1}^{n} 
\left(\frac{\partial A}{\partial q_{\mbox{\tiny(k)}}}
\frac{\partial B}{\partial p_{\mbox{\tiny(k)}}}
 - \frac{\partial A}{\partial p_{\mbox{\tiny(k)}}}
 