Define $f^{\prime}:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$ by $f^{\prime}(z_{1},z_{2},z_{3})=(a,\mathop{\rm Re}c,\mathop{\rm Im}c)$, where

	$\displaystyle a$	$\displaystyle=|z_{1}|^{2}-|z_{2}|^{2}$		(27)
	$\displaystyle\text{and}\quad c$	$\displaystyle=\begin{cases}z_{3}+\bar{z}_{1}\bar{z}_{2}/|z_{1}|,&\text{$a=0$ and\/ $z_{1},z_{2}\neq 0$,}\\ z_{3},&\text{$a=z_{1}=z_{2}=0$,}\\ z_{3}+\bar{z}_{1}\bar{z}_{2}/|z_{1}|,&\text{$a>0$,}\\ z_{3}+\bar{z}_{1}\bar{z}_{2}/|z_{2}|,&\text{$a<0$.}\end{cases}$		(28)

Then $f^{\prime}$ is continuous and piecewise smooth, and $(f^{\prime})^{-1}(a,\mathop{\rm Re}c,\mathop{\rm Im}c)=N^{\prime}_{a,c}$, where $N^{\prime}_{a,c}$ is given in Definition 5. Hence, $f^{\prime}$ is a piecewise smooth special Lagrangian fibration of $\mathbin{\mathbb{C}}^{3}$.