Example 6.9 Let $\alpha,b,c\in\mathbin{\mathbb{R}}$ and define $u(x,y)=\alpha y+b$ and $v(x,y)=\alpha x+c$. Then $u,v$ satisfy (33) for any value of $a$. The corresponding special Lagrangian 3-folds are the result of applying a diagonal $\mathop{\rm SU}(3)$ matrix to one of the fibres of the fibration of Corollary 4.2.

Example 6.10 Define $u(x,y)={\textstyle\frac{1}{2}}y^{2}{\textstyle\mathop{\rm sech}}^{2}x-{\textstyle\frac{1}{2}}\cosh^{2}x$ and $v(x,y)=y\tanh x$. Then $u$ and $v$ satisfy (32). Equation (31) with $a=0$ defines an explicit nonsingular special Lagrangian 3-fold $N$ in $\mathbin{\mathbb{C}}^{3}$. It can be shown that $N$ is ruled, and arises from Harvey and Lawson’s ‘austere submanifold’ construction [9, §III.3.C] of SL $m$-folds in $\mathbin{\mathbb{C}}^{m}$, as the normal bundle of a catenoid in $\mathbin{\mathbb{R}}^{3}$.

Example 6.11 Define $u(x,y)=-y\sinh 2x$ and $v=y-{\textstyle\frac{1}{2}}\cosh 2x$ on the half-plane $y\geqslant 0$ in $\mathbin{\mathbb{R}}^{2}$. Then $u,v$ satisfy (32). So equation (31), with the additional condition that $y=\mathop{\rm Im}(z_{1}z_{2})\geqslant 0$, defines an explicit special Lagrangian 3-fold $N$ in $\mathbin{\mathbb{C}}^{3}$. It turns out (surprisingly) that $N$ is nonsingular, and is equivalent to one of the SL 3-folds constructed in [12, Ex. 7.4] by evolving paraboloids in $\mathbin{\mathbb{C}}^{3}$.