an almost holomorphic extension of $\widetilde{p}$ and especially the real part of that function. The general form of a smooth I-Lagrangian manifold $\Lambda $, for which the $x$-space projection $\Lambda\ni (x,\xi ) \mapsto x\in{\bf
  C}^n$ is a local diffeomorphism, is locally $\Lambda =\Lambda _\Phi $ where $\Phi $ is real and smooth and we define $$\Lambda _\Phi = \{ (x,\frac{2}{i}\frac{\partial \Phi }{\partial x});\, x\in \Omega \} ,\ \Omega \subset {\bf C}^n\hbox{ open}.$$ With a slight abuse of notation, we can therefore identify the ${\bf
  C}$-Lagrangian manifold $\Lambda _{\phi _0}$ with the I-Lagrangian manifold $\Lambda _{-\Im \phi _0}$, since for holomorphic functions (or more generally where $\overline{\partial }_x\phi _0=0$, we have $\frac{\partial \phi _0}{\partial x}=\frac{2}{i}\frac{\partial -\Im
  \phi _0}{\partial x}$.

(\[ev.0.4\]) shows that $$\Phi _0(x)+\Phi _0(\overline{\theta } )-(-\Im \phi _0 (\cdot ,\overline{\theta} ))\asymp
|x-\overline{\theta }|^2.$$ Thus, if we define [$$\label{ev.0.16}
\Lambda _{\Phi _t}=\exp (tH_{\Re \widetilde{p}}^{-\Im \sigma })(\Lambda
_{\Phi _0}),$$]{} and fix the $t$-dependent constant in this defintition of $\Phi _t$ by imposing the real Hamilton-Jacobi equation, [$$\label{ev.0.17}
\partial _t\Phi _t+\Re \widetilde{p}(x,\frac{2}{i}\frac{\partial \Phi
  _t}{\partial x})=0,\ \Phi _{t=0}=\Phi _0,$$]{} and noticing that the real part of (\[ev.0.8\]) is a similar equation for $-\Im \phi _t$, [$$\label{ev.0.18}
\partial _t(-\Im \phi
)+\Re \widetilde{p}(x,\frac{2}{i}\frac{\partial }{\partial
  x}(-\Im \phi 