the complex domain, so we cannot hope to solve the eiconal equation exactly, but we can do so to infinite order at $t=0$, $x=\overline{y}=\theta $. If we put [$$\label{ev.0.10}
\Lambda _{\phi _t(\cdot ,\theta )}=\{ (x,\phi '_x(t,x,\theta ))\},$$]{} we have to $\infty $ order at $t=0$, $\theta =x$: [$$\label{ev.0.11}
\Lambda _{\phi _t(\cdot ,\theta )}=\exp (t\widehat{H}_{\frac{1}{i}\widetilde{p}})
(\Lambda _{\phi _0(\cdot ,\theta )}).$$]{} with $\widehat{H}_{\widetilde{p}}=H_{\widetilde{p}}+\overline{H_{\widetilde{p}}}$ denoting the real vector field associated to the (1,0)-field $H_p$, and similarly for $\widehat{H}_{\frac{1}{i}\widetilde{p}}$. (We sometimes neglect the hat when integrating the Hamilton flows.) At a point where $\overline{\partial }\widetilde{p}=0$, we have [$$\label{ev.0.12}
\widehat{H}_{\widetilde{p}}=H^{\Re \sigma }_{\Re\widetilde{p}}=H^{\Im\sigma
}_{\Im\widetilde{p}},\quad \widehat{H}_{i\widetilde{p}}=
-H^{\Re\sigma }_{\Im\widetilde{p}}=H^{\Im \sigma }_{\Re\widetilde{p}},$$]{} where the other fields are the Hamilton fields of $\Re \widetilde{p}$, $\Im \widetilde{p}$ with respect to the real symplectic forms $\Re
\sigma $ and $\Im \sigma $ respectively. See [@Sj82; @MeSj02]. Thus (\[ev.0.11\]) can be written [$$\label{ev.0.13}
\Lambda _{\phi _t(\cdot ,\theta )}=\exp (tH^{-\Im \sigma }_{\Re
  \widetilde{p}})(\Lambda _{\phi _0(\cdot ,\theta )}).$$]{} A complex Lagrangian manifold is also an I-Lagrangian manifold (i.e. a Lagrangian manifold for $\Im \sigma $) so (\[ev.0.13\]) can be viewed as a relation between I-Lagrangian manifolds and it defines the I-Lagrangian manifold $\Lambda _{\phi _t(\cdot ,\theta )}$ in an unambigious way, once we have fixed 