On normal forms of singular Levi-flat real analytic hypersurfaces

Let F(z) = Re(P(z)) + h.o.t be such that M = (F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂn, n ≥ 2, where P (z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂn and Milnor number µ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ(M) = (Re(h) = 0), where h(z) is a polynomial of degree µ + 1 and j0k (h) = P.