Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,\omega)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $\omega$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^\omega\subset T^*M$ be the canonical symplectization of $(M,\omega)$ and $M$ be identified with the zero section of $M^\omega$. Then $M^\omega$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+i\sigma\mapsto \phi_p(t+i\sigma)=\sigma\omega_{g_t(p)}$, where $p\inM$ is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $\omega$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^\omega$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^\omega$ if all the parametrizations $\phi_p(t+i\sigma)$ defined above are holomorphic on $\phi_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^\omega$, then the real function $E$ on $M^\omega\subset T^*M$ defined by the condition $\alpha=E(\alpha)\omega_{\pi(\alpha)}$, for each $\alpha\in M^\omega$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Amp\`ere equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $\omega$ are real analytic then the symplectization $M^\omega$ admits an unique maximal adapted complex tube.