On Normal Forms of Singular Levi-Flat Real Analytic Hypersurfaces

Let $F(z)=\mathcal{R}e(P(z))+h\mathrm{.}o\mathrm{.}t$ be such that $M=(F=0)$ defines a germ of real analytic Levi-flat at $0\in {\mathbb{C}}^n$, $n\ge 2$, where $P(z)$ is a homogeneous polynomial of degree $k$ with an isolated singularity at $0\in {\mathbb{C}}^n$ and Milnor number $\mu$. We prove that there exists a holomorphic change of coordinate $\varphi$ such that $\varphi (M)=(\mathcal{R}e(h)=0)$ where $h(z)$ is a polynomial of degree $\mu +1$ and $j_0^k(h)=P$.