y,\eta $ becomes $$\left(\begin{array}{ccc}-\Im \phi ''_{y,y} &B &0\\
{{^t\hskip -2pt
B}} &C &-1\\
0 &-1 &0
 \end{array}\right)$$ which is nondegenerate independently of $B,C$.

If $\Phi (x)$ denotes the critical value in (\[ir.2\]), it remains to check that $\frac{2}{i}\frac{\partial \Phi }{\partial
  x}=\xi $ where $\xi =\frac{\partial \phi }{\partial
  x}(x,\widetilde{y})$, $(\widetilde{y},\eta )$ denoting the critical point. However, since $\Phi $ is a critical value, we get $$\frac{2}{i}\frac{\partial \Phi }{\partial x}=\frac{2}{i}\frac{\partial
}{\partial x}(-\Im \phi (x,\widetilde{y}))=\frac{\partial \phi
}{\partial x}(x,\widetilde{y}).$$

Also notice that when $G=0$, the formula (\[ir.2\]) produces the same function as (\[ir.1\]).

Write $\widetilde{y}=y+i\theta $ and consider the function [$$\label{ir.5}
f(x;y,\eta ;\theta )=-\Im \phi (x,y+i\theta )-\eta \cdot \theta ,$$]{} which appears in (\[ir.2\]).

\[ir2\] $f$ is a nondegenerate phase function with $\theta $ as fiber variables which generates a canonical transformation which can be identified with $\kappa _T$.

$$\frac{\partial }{\partial \theta }f=-\Re \frac{\partial \phi
}{\partial \widetilde{y}}(x,y+i\theta )-\eta ,$$ so $f$ is nondegenerate. The canonical relation has the graph $$\begin{aligned}
&\{ (x,\frac{\partial \phi }{\partial x};y,\eta ,\frac{\partial
}{\partial y}\Im \phi (x,y+i\theta ),\theta );\ \eta =-\Re
\frac{\partial \phi }{\partial \widetilde{y}}(x,y+i\theta )\}&\\
&=\{ (x,\frac{\partial \phi }{\partial x}(x,y+i\theta );y,-\Re
\frac{\partial \phi }{\partial \widetilde{y}}(x,y+i\theta ),\Im
\frac{\partial \phi }{\partial \widetilde{y}}(x,y+i\theta ),\theta )\}&,\end{aligned}$$ and up to reshuffling of the components on 