theory of [@HeSj86].

We shall work locally. Let $G(y,\eta )\in C^\infty
(\mathrm{neigh\,}((y_0,\eta _0),{\bf R}^{2n}))$ be real-valued and small in the $C^{\infty }$ topology. Then $$\Lambda _G=\{ (y,\eta )+iH_G(y,\eta );\, (y,\eta )\in \mathrm{neigh\,}((y_0,\eta _0))\},\ H_G=\frac{\partial G}{\partial \eta }\frac{\partial }{\partial y
}-
\frac{\partial G}{\partial y}\frac{\partial }{\partial \eta }$$ is an $I$-Lagrangian manifold, i.e. a Lagrangian manifold for the real symplectic form $\Im \sigma $, where $\sigma $ denotes the complex symplectic form $\sum_1^n d\widetilde{\eta }_j\wedge
d\widetilde{y}_j$. Here, for notational reasons we reserve the notation $(y,\eta )$ for the real cotangent variables and let the tilde indicate that we take the corresponding complexified variables.

We may also represent $\Lambda _G$ by means of a nondegenerate phase function in the sense of Hörmander in the following way:

Consider $$\psi (\widetilde{y},\eta )=-\eta \cdot \Im \widetilde{y}+G(\Re
\widetilde{y},\eta )$$ where $\widetilde{y}$ is complex and $\eta $ real according to the convention above. Then $$\nabla _\eta \psi (\widetilde{y},\eta )=-\Im \widetilde{y}+\nabla
_\eta G(\Re \widetilde{y},\eta ),$$ and since $G$ is small, we see that $d\frac{\partial \psi }{\partial
  \eta _1},...,d\frac{\partial \psi }{\partial \eta _n}$ are linearly independent. So $\psi $ is indeed a nondegenerate phase function if we drop the classical requirement of homogeneity in the $\eta $ variables.

Let $$C_\psi 