The purpose of the paper is to give a geometric criterion which guarantees that the automorphism groups of certain domains \(\Omega \subset\mathbb C^2\) only contain maps whose components each depend on only one variable and in at least one variable this dependence is the identity. There are two versions of this, one for real analytic boundary and one for domains with Levi-flat boundary. First for a domain \(\Omega\subset\mathbb C^2\) with real analytic boundary, let \(T_0^{\mathbb C}\) denote its complex tangent space at the origin (assumed to be in \(\partial\Omega\)). Assume that:

(1)

the set \( T_0^{\mathbb C}\cap\Omega\) is nonempty and open in \(T_0^{\mathbb C}\);

(2)

\(\Omega\) is defined by \(\text{Re}\;w + \eta(z,\overline{z}) + \psi(z,\overline{z},\text{Im}\;w) \cdot (\text{Im}\;w)^2 < 0\) in coordinates \((z,w)\) and \(\eta\) has no harmonic term;

(3)

for \(f\in \text{Aut}_0(\Omega)\), the function \((z,w)\mapsto (f(z,0),w)\) extends to \(\Omega\) as an automorphism fixing 0;

(4)

0 is not a boundary accumulation point.

Then every automorphism of \(\Omega\) fixing 0 has the form \((z,w)\mapsto (\varphi(z),w)\).
The Levi-flat case assumes that a local defining function for \(\Omega\) has the form \(\text{Re}\;w=0\), assumption (3) above, and every automorphism of \(\Omega\) extends holomorphically to some neighbourhood of 0. The conclusion is then the same.