i})}}}, 
z_{t^i}^{\mbox{\tiny{({\it i})}}}\}_{\mbox{\tiny{PB}}}&=& \nonumber \\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\sum_{a^i,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}_{t^i-1}^{\mbox{\tiny{({\it i})}}},
\tilde{G}_{t^i}^{\mbox{\tiny{({\it i})}}}, 
\tilde{G}_{t^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_{a^i}^{\mbox{\tiny{({\it i})}}}, 
z_{s^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{t^i-1}^{\mbox{\tiny{({\it i})}}}, 
x_{b^i}^{\mbox{\tiny{({\it i})}}}, 
z_{t^i+1}^{\mbox{\tiny{({\it i})}}}, \cdots,
z_{m^i}^{\mbox{\tiny{({\it i})}}})}~,~~
\label{phi-2n+m6}\end{aligned}$$ where $s^i< t^i$. We refer to the systems where the master equations are given by Eq. (\[N-eqf(m)\]) or Eq. (\[N-eqf(m)2\]) as generalized Nambu systems.

Nambu systems in constrained Hamiltonian systems {#Nambu systems in constrained Hamiltonian systems}
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Subject {#Subject}
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In the previous section, we found that a Hamiltonian system can be formulated as a Nambu system with multiplets including composite variables of $q$ and $p$. The main feature of our formulation is the existence of induced constraints that are required just for consistency between the variables. Together with the Hamiltonian of the original system, the induced constraints serve as Hamiltonians of the Nambu system. Therefore it is intriguing to study how constrained Hamiltonian systems, systems with [*physical*]{} constraints, are cast into Nambu systems in our formulation.

The relations between Nambu systems and constrained Hamiltonian systems have been investigated by many authors [@BF; @CK; @R; @MS; @KT; @KT2; @LJ]. To clarify the 