On the dimension of the group of automorphisms of an analytic hypersurface

Let $M$ be a nondegenerate real analytic hypersurface in $\mathbf C^2$, let $\xi\in M$, and let $G_\xi$ consist of the automorphisms of $M$ fixing the point $\xi$. Then, as follows from a theorem of Moser, the real dimension of $G_\xi$ does not exceed 5. Here it is shown that 1) dimensions 2, 3, and 4 cannot be realized, but for 0, 1, and 5 examples are given; 2) if the point $\xi$ is not umbilical, then $G_\xi$ consists of not more than two mappings.
Bibliography: 4 titles.