=\{ (\widetilde{y},\eta )\in \mathrm{neigh\,}((y_0,\eta _0),{\bf C}^n\times {\bf R}^n);\, \nabla _\eta \psi =0\}$$ and consider the corresponding I-Lagrangian manifold $$\Lambda _\psi =\{ (\widetilde{y},\frac{2}{i}\frac{\partial \psi
}{\partial \widetilde{y}}(\widetilde{y},\eta ));\, (\widetilde{y},\eta
)\in C_\psi \} .$$ Here we adopt the convention that $\frac{\partial }{\partial
  \widetilde{y}}$ denotes the holomorphic derivative, since $\widetilde{y}$ are complex variables: $$\frac{\partial }{\partial \widetilde{y}}=\frac{1}{2}(\frac{\partial
}{\Re \widetilde{y}}+\frac{1}{i}\frac{\partial }{\partial \Im \widetilde{y}}).$$ Let us first check that that $\Lambda _\psi $ is I-Lagrangian, using only that $\psi $ is a nondegenerate phase function: That $\Lambda _\psi $ is a submanifold with the correct real dimension $=2n$ is classical since we can identify $\frac{2}{i}\frac{\partial \psi
}{\partial \widetilde{y}}$ with $\nabla _{\Re \widetilde{y}, \Im
  \widetilde{y}}\psi $. Further, $$\begin{aligned}
&&-\Im {{(\widetilde{\eta }\cdot d\widetilde{y})}_\vert}_{\Lambda
  _\psi }\simeq -\Im {{(\frac{2}{i}\frac{\partial \psi }{\partial
  \widetilde{y}}\cdot d\widetilde{y})}_\vert}_{C_{\psi }}=\\
&&-\frac{1}{2i}{{(\frac{2}{i}\frac{\partial \psi }{\partial
      \widetilde{y}}d\widetilde{y}+\frac{2}{i}\frac{\partial \psi
    }{\partial
      \overline{\widetilde{y}}}d\overline{\widetilde{y}})}_\vert}_{C_\psi }=
{{(\frac{\partial \psi }{\partial
      \widetilde{y}}d\widetilde{y}+\frac{\partial \psi }{\partial
      \overline{\widetilde{y}}}d\overline{\widetilde{y}})}_\vert}_{C_\psi}\\
&&={{d\psi }_\vert}_{C_\psi }\end{aligned}$$ which is a closed form and using that $\Im \sigma =d\Im (\widetilde{\eta
}\cdot d\widetilde{y})$, we get $$-{{\Im \sigma }_\vert}_{\Lambda _\psi }=0.$$

We next check for our specific phase $\psi $ that $\Lambda _{\psi
  }=\Lambda _G$: If $(\widetilde{y},\frac{2}{i}\frac{\partial \psi
  }{\partial \widetilde{y}}(\widetilde{y},\eta ))$ is a general point on $\Lambda _\psi $, then $\Im \widetilde{y}=\nabla _\eta G(\Re
  \widetilde{y},\eta )$ and $$\begin{aligned}
\frac{2}{i}\frac{\partial 