Definition 5.1 Let $a\in\mathbin{\mathbb{R}}$ and $c\in\mathbin{\mathbb{C}}$. Define a special Lagrangian 3-fold $N_{a,c}$ in $\mathbin{\mathbb{C}}^{3}$ as follows:

• (i)

  When $a=0$, define

  $$\begin{split}N_{0,c}=\Bigl\{(z_{1}&,z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:|z_{1}|^{2}=|z_{2}|^{2}=|z_{3}-c|^{2},\\ &\mathop{\rm Im}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)=0,\quad\mathop{\rm Re}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)\geqslant 0\Bigr\}.\end{split}$$

  (21)

  Then $N_{0,c}$ is the translation of the special Lagrangian $T^{2}$-cone $L_{0}^{+}$ of (16) by the vector $(0,0,c)$. It has one singular point at $(0,0,c)$.

• (ii)

  When $a>0$, define

  $$\begin{split}N_{a,c}=\Bigl\{(z_{1}&,z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:|z_{1}|^{2}-a=|z_{2}|^{2}=|z_{3}-c|^{2},\\ &\mathop{\rm Im}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)=0,\quad\mathop{\rm Re}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)\geqslant 0\Bigr\}.\end{split}$$

  (22)

  Then $N_{a,c}$ is the translation of the nonsingular SL 3-fold $L_{1,a}^{+}$ of (17) by the vector $(0,0,c)$. It is diffeomorphic to ${\mathcal{S}}^{1}\times\mathbin{\mathbb{R}}^{2}$.

• (iii)

  When $a<0$, define

  $$\begin{split}N_{a,c}=\Bigl\{(z_{1}&,z_{2},z_{3})\in\mathbin{\mathbb{C}}^{3}:|z_{1}|^{2}=|z_{2}|^{2}+a=|z_{3}-c|^{2},\\ &\mathop{\rm Im}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)=0,\quad\mathop{\rm Re}\bigl(z_{1}z_{2}(z_{3}-c)\bigr)\geqslant 0\Bigr\}.\end{split}$$

  (23)

  Then $N_{a,c}$ is the translation of the nonsingular SL 3-fold $L_{2,-a}^{+}$ of (18) by the vector $(0,0,c)$. It is diffeomorphic to ${\mathcal{S}}^{1}\times\mathbin{\mathbb{R}}^{2}$.