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

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

(8)

Then $K_{a,b,c}$ is a special Lagrangian $3$-fold in $\mathbin{\mathbb{C}}^{3}$. If $a,b$ are not both zero, then $K_{a,b,c}$ is a nonsingular embedded submanifold diffeomorphic to ${\mathcal{S}}^{1}\times\mathbin{\mathbb{R}}^{2}$. Also $K_{0,0,c}$ is the union of the two special Lagrangian $3$-planes

$$\begin{split}\Pi^{+}_{c}&=\bigl\{(z,i\bar{z},t+ic):z\in\mathbin{\mathbb{C}},\quad t\in\mathbin{\mathbb{R}}\bigr\}\\ \text{and}\qquad\Pi^{-}_{c}&=\bigl\{(z,-i\bar{z},t+ic):z\in\mathbin{\mathbb{C}},\quad t\in\mathbin{\mathbb{R}}\bigr\},\end{split}$$

(9)

which intersect in the real line $\bigl\{(0,0,t+ic):t\in\mathbin{\mathbb{R}}\bigr\}$. It is singular as an embedded submanifold, but nonsingular as an immersed submanifold.