My question is of local nature.  

Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.  

Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)$$ is a CR manifold. Let me assume that $M$ is the simplest possible kind of CR manifold, namely that it is foliated by real-codimension-one complex submanifolds.


[Equivalently, for those who don't know what CR manifolds are, consider the hyperplane distribution $L:=TM\cap i\cdot TM\subset TM$. I require the distribution $L$ to be integrable, i.e., to come from a (real codimension $1$) foliation of $M$.]



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> Under the above assumptions, is $f^{-1}\big([0,\infty)\big)$ locally isomorphic to $$\big\{(z\_1,...z\_n)\in\mathbb C^n\,:\,\mathrm{im}(z\_1)\ge 0\big\}?$$
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I.e., does there exist a neighbourhood $U\subset f^{-1}([0,\infty))$ of zero and an isomorphism $\varphi:U\to \big\{z\in\mathbb C^n\,:\,\sum|z\_i|^2<1,\,\mathrm{im}(z\_1)\ge 0\big\}$ which is holomorphic in the interior and smooth all the way to the boundary.



Complex manifold with boundary
Perhaps Giuseppe Della Sala's paper might be useful here: <https://www.ams.org/journals/proc/2011-139-07/S0002-9939-2010-10746-3/home.html>


It precisely deals with the equivalence of smooth Levi-flats. There are examples in the paper


