difference between previous works and our approach, here we give a brief summary of the results obtained so far. In most works, Nambu systems are treated as the original systems, and studies have been carried out to find appropriate constrained Hamiltonian systems into which the Nambu systems can be embedded [@BF; @CK; @R; @MS; @KT; @KT2]. Specifically, it has been shown that Nambu equations (Eq. (\[N-eq\])) are compatible with the following equations: $$\begin{aligned}
&& p_i = H_1 \frac{\partial H_2}{\partial x_i}~,\label{N-eq-con1}\\
&& \sum_{i=1}^3 \frac{\partial(H_1, H_2)}{\partial(x_i, x_j)}\frac{dx_i}{dt} = 0~.
\label{N-eq-con2}\end{aligned}$$ Here $p_i$ ($i=1,2,3$) are the canonical conjugate momenta defined as $p_i \equiv \partial L/\partial \dot{x}_i$ with the Lagrangian $$\begin{aligned}
L = H_1 \sum_{i=1}^3 \frac{\partial H_2}{\partial x_i}\frac{dx_i}{dt} ~.
\label{N-eq-conL}\end{aligned}$$ Equation (\[N-eq-con2\]) can be derived as the Euler–Lagrange equation from this Lagrangian, and Eq. (\[N-eq-con1\]) leads to the relations $\phi_i \equiv p_i - H_1 {\partial H_2}/{\partial x_i} = 0$, which can be regarded as constraints. In this way, Nambu systems can be interpreted as Hamiltonian systems with specific constraints.

On the other hand, researchers have studied whether constrained systems can be described as Nambu systems or not. Specifically, it has been shown that constrained Hamiltonian systems can be formulated in terms of (a generalized 