Let $a,b,c\in\mathbin{\mathbb{R}}$, and define a subset $L_{a,b,c}$ in $\mathbin{\mathbb{C}}^{3}$ by

$$\begin{split}L_{a,b,c}=\bigl\{(z_{1},z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:\,&|z_{1}|^{2}-|z_{2}|^{2}=a,\\ &|z_{1}|^{2}-|z_{3}|^{2}=b,\quad\mathop{\rm Im}(z_{1}z_{2}z_{3})=c\bigr\}.\end{split}$$

(12)

Then $L_{a,b,c}$ is a special Lagrangian $3$-fold in $\mathbin{\mathbb{C}}^{3}$. Moreover

• (i)

  $L_{0,0,0}$ has one singular point at $0$.

• (ii)

  If $\alpha>0$ then $L_{\alpha,\alpha,0}$ has singular set $\bigl\{(\alpha^{1/2}{\rm e}^{i\theta},0,0):\theta\in[0,2\pi)\bigr\}$.

• (iii)

  If $\alpha>0$ then $L_{-\alpha,0,0}$ has singular set $\bigl\{(0,\alpha^{1/2}{\rm e}^{i\theta},0):\theta\in[0,2\pi)\bigr\}$.

• (iv)

  If $\alpha>0$ then $L_{0,-\alpha,0}$ has singular set $\bigl\{(0,0,\alpha^{1/2}{\rm e}^{i\theta}):\theta\in[0,2\pi)\bigr\}$.

All other $L_{a,b,c}$ are nonsingular embedded submanifolds diffeomorphic to $T^{2}\times\mathbin{\mathbb{R}}$.