form of) Nambu mechanics by introducing an extra phase-space variable [@LJ]. For a system with canonical variables $(q_k, p_k)$ $(k=1, \cdots, n)$ and $m$ first class constraints $\phi_l(q_1, \cdots, p_n)=0$, the equations of motion are given by $$\begin{aligned}
\frac{d q_k}{d t} &=& \frac{\partial H}{\partial p_k} 
+ \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial p_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial p_k}
\right)~,\label{eq-constraints1}\\~~
\frac{d p_k}{d t} &=& -\frac{\partial H}{\partial q_k} 
- \sum_{l=1}^m \left(\
\frac{\partial \lambda_l}{\partial q_k}\phi_l
+ \lambda_l \frac{\partial \phi_l}{\partial q_k}
\right)~,
\label{eq-constraints2}\end{aligned}$$ where $\lambda_l$ are Lagrange multipliers. Equations (\[eq-constraints1\]) and (\[eq-constraints2\]) are derived from (a generalized form of) the Nambu equation $$\begin{aligned}
\frac{d f}{dt} = \sum_{k=1}^n \frac{\partial (f, H_1, H_2)}{\partial (q_k, p_k, r)}~,
\label{N-eq-constraints}\end{aligned}$$ where $f=f(q_1, \cdots, p_n)$, $r$ is an extra phase-space variable, and Hamiltonians are defined as $$\begin{aligned}
H_1 = H - r~,~~ H_2 = r+ \sum_{l=1}^m \lambda_l \phi_l~.
\label{H1H2-constraints}\end{aligned}$$ The equation for $r$ is given by $$\begin{aligned}
\frac{d r}{dt}= - \sum_{l=1}^m 
\left(\lambda_l \{\phi_l, H\}_{\mbox{\tiny{PB}}}
+\phi_l \{\lambda_l, H\}_{\mbox{\tiny{PB}}}
\right)
= - \sum_{l=1}^m \lambda_l \frac{d \phi_l}{dt},
\label{r-eq-constraints}\end{aligned}$$ where the last equality holds after imposing constraints. Requiring the extra variable $r$ to decouple from the dynamics, i.e., $dr/dt = 0$, we obtain ${d \phi_l}/{dt} = 0$.

Our approach differs from these previous works. Our starting 