Assumption 7.2 Suppose that each fibre $N_{a,b,c}$ of $f$ may be written

$$\begin{split}N_{a,b,c}=\Bigl\{(&z_{1},z_{2},z_{3})\in U:\mathop{\rm Re}(z_{1}z_{2})=u_{a,b}\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr),\\ &\mathop{\rm Im}(z_{3})=v_{a,b}\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr)+c,\;\>|z_{1}|^{2}-|z_{2}|^{2}=a\Bigr\},\end{split}$$

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where $u_{a,b},v_{a,b}:V_{a,b}\rightarrow\mathbin{\mathbb{R}}$ are 2-parameter families of functions and

$$V_{a,b}=\bigl\{\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr):(z_{1},z_{2},z_{3})\in N_{a,b,0}\bigr\}$$

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is an open set in $\mathbin{\mathbb{R}}^{2}$. Suppose also that the $u_{a,b},v_{a,b}$ satisfy:

• (i)

  $u_{0,b},v_{0,b}$ are smooth except at points $(x,0)$ in $V_{0,b}$ with $u_{0,b}(x,0)=0$, and

  $$\frac{\partial u_{0,b}}{\partial x}=-2\bigl(u_{0,b}^{2}+y^{2}\bigr)^{1/2}\frac{\partial v_{0,b}}{\partial y}\quad\text{and}\quad\frac{\partial u_{0,b}}{\partial y}=\frac{\partial v_{0,b}}{\partial x}$$

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  hold except at these points.

• (ii)

  When $a\neq 0$, $u_{a,b}$ and $v_{a,b}$ are smooth on $V_{a,b}$ and satisfy

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

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• (iii)

  $u_{a,b}$ and $v_{a,b}$ depend continuously on $a,b$, and smoothly wherever $a\neq 0$.

• (iv)

  $u_{a,b}\equiv u_{-a,b}$ and $v_{a,b}\equiv v_{-a,b}$ for all $a,b$.

• (v)

  $u_{a,b}(x,-y)=u_{a,b}(x,y)$ and $v_{a,b}(x,-y)=-v_{a,b}(x,y)$ for all $a,b,x,y$.

• (vi)

  $u_{a,b}(-x,y)=u_{a,b}(x,y)$ and $v_{a,b}(-x,y)=-v_{a,b}(x,y)$ for all $a,b,x,y$.