Lemma 6.2 [03LH]

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Lemma 6.2

Let ${\bf z}=(z_{1},z_{1},z_{3})\in N$ , with $z_{1}\neq 0$ . Set $x=\mathop{\rm Re}(z_{3})$ and $y=\mathop{\rm Im}(z_{1}^{2})$ . Then $N$ is nonsingular at $\bf z$ , and $T_{\bf z}N=\langle{\bf p}_{1},{\bf p}_{2},{\bf p}_{3}\rangle_{\scriptscriptstyle\mathbb{R}},$ where

$\displaystyle{\bf p}_{1}$	$\displaystyle=(iz_{1},-iz_{1},0),$	(34)
$\displaystyle{\bf p}_{2}$	$\displaystyle=\bigl((2z_{1})^{-1}{\textstyle\frac{\partial u}{\partial x}}(x,y),(2z_{1})^{-1}{\textstyle\frac{\partial u}{\partial x}}(x,y),1+i{\textstyle\frac{\partial v}{\partial x}}(x,y)\bigr)\quad\text{and}$	(35)
$\displaystyle{\bf p}_{3}$	$\displaystyle=\bigl((2z_{1})^{-1}({\textstyle\frac{\partial u}{\partial y}}(x,y)+i),(2z_{1})^{-1}({\textstyle\frac{\partial u}{\partial y}}(x,y)+i),i{\textstyle\frac{\partial v}{\partial y}}(x,y)\bigr).$	(36)