Let $u,v:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$ be continuous, and let $a\in\mathbin{\mathbb{R}}$. Define

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

(31)

Then

• (a)

  If $a=0$, then $N$ is a singular special Lagrangian $3$-fold in $\mathbin{\mathbb{C}}^{3}$ if $u,v$ are differentiable and satisfy

  $$\frac{\partial u}{\partial x}=-2\bigl(u^{2}+y^{2}\bigr)^{1/2}\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x},$$

  (32)

  except at points $(x,0)$ in $\mathbin{\mathbb{R}}^{2}$ with $u(x,0)=0$, where $u,v$ need not be differentiable. The singular points of $N$ are those of the form $(0,0,z_{3})$, where $z_{3}=x+iv(x,0)$ for $x\in\mathbin{\mathbb{R}}$ with $u(x,0)=0$.

• (b)

  If $a\neq 0$, then $N$ is a nonsingular special Lagrangian $3$-fold in $\mathbin{\mathbb{C}}^{3}$ if and only if $u,v$ are differentiable on all of $\mathbin{\mathbb{R}}^{2}$ and satisfy

  $$\frac{\partial u}{\partial x}=-\bigl(4u^{2}+4y^{2}+a^{2}\bigr)^{1/2}\frac{\partial v}{\partial y}\quad\text{and}\quad\frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}.$$

  (33)