The author shows the following:

a) If $M$ is a real analytic surface in ${\u2102}^{3}$ with a non-degenerate complex tangent at $p$, then there is a local biholomorphic map $\phi $ of ${\u2102}^{3}$, $\phi \left(p\right)=0$, such that $\phi \left(M\right)$ is given by ${z}_{2}={\overline{z}}_{1}^{2}$, ${z}_{3}={z}_{1}{\overline{z}}_{1}$.

b) If $M$ is a real analytic submanifold of ${\u2102}^{5}$ of dimension 4 with a non-degenerate complex tangent at $p$, then there is a local biholomorphic map $\phi $ of ${\u2102}^{5}$, $\phi \left(p\right)=0$, such that $\phi \left(M\right)$ is given by ${z}_{5}={z}_{1}({\overline{z}}_{1}+{x}_{2}+i{x}_{3})$, ${z}_{4}={({\overline{z}}_{1}+{x}_{2}+i{x}_{3})}^{2}$, ${y}_{2}={y}_{3}=0$.

The proofs of both theorems involve a rapid iteration argument, by solving a linearized functional equation first. The second normal form is more difficult because of the non-trivial quadratic terms. To avoid the radius of convergence of the linearized functional equation shrinking too much, the author has to find a good solution among all possible ones. This is the main novelty of the paper.

The normal form of real analytic surfaces in ${\u2102}^{2}$ has been studied by *J. K. Moser* and *S. M. Webster* [Acta Math. 150, 255–296 (1983; Zbl 0519.32015)].