Assumption 7.1 Let $U$ be a connected open neighbourhood of $(0,0,0)$ in $\mathbin{\mathbb{C}}^{3}$, which is invariant under the symmetries in parts (ii) and (v)–(viii) below. We aim to construct a special Lagrangian fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$, with fibres $N_{a,b,c}=f^{-1}(a,b,c)$, with the following properties:

• (i)

  $f$ is continuous, and smooth except on the real hypersurface $|z_{1}|=|z_{2}|$.

• (ii)

  $f({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3})=f(z_{1},z_{2},z_{3})$ for all $(z_{1},z_{2},z_{3})\in U$ and $\theta\in\mathbin{\mathbb{R}}$. Equivalently, every fibre $N_{a,b,c}$ is invariant under the $\mathbin{\rm U}(1)$-action given by

  $${\rm e}^{i\theta}:(z_{1},z_{2},z_{3})\mapsto({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3}).$$

  (52)

• (iii)

  If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $a=|z_{1}|^{2}-|z_{2}|^{2}$.

• (iv)

  The set of singular points of singular fibres of $f$ is $\bigl\{(0,0,z_{3})\in U\bigr\}$. In particular, $N_{a,b,c}$ is nonsingular if $a\neq 0$.

• (v)

  If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{1},z_{2},z_{3}+it)=(a,b,c+t)$ for all $t\in\mathbin{\mathbb{R}}$. This means that $N_{a,b,c+t}$ is the translation of $N_{a,b,c}$ by $(0,0,it)$, and that

  $$N_{a,b,c}=\bigl\{(z_{1},z_{2},z_{3}+ic):(z_{1},z_{2},z_{3})\in N_{a,b,0}\bigr\}.$$

  (53)

• (vi)

  If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{2},z_{1},z_{3})=(-a,b,c)$.

• (vii)

  If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(\bar{z}_{1},\bar{z}_{2},\bar{z}_{3})=(a,b,-c)$.

• (viii)

  If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{1},z_{2},-z_{3})=(a,b,-c)$.