ИСТИНА

In CR geometry one usually considers manifolds with constant CR dimension (i.e., the dimension of complex tangent space). If this dimension becomes bigger at some point, then we call such point RC-singular point. All the objects are considered up to biholomorphic transformations. The author is going to speak about his recent results, connected with the study of holomorphic symmetries of real 2-dimensional submanifolds of ${\bf C}^2$ in RC-singular points. As a measure of symmetricity of a germ $M_{\xi}$ we use such traditional characteristic as the dimension of Lie algebra ${\rm aut} \,M_{\xi}$ of germs of tangent to $M_{\xi}$ at point $\xi$ vector fields. The following result is obtained. Let $$Q_{0} =\{(z,w) \in {\bf C}^2 : w=z^m \, \bar{z}^n \}, \, n,m\in {\bf N}, $$ be a 2-dimensional surface with RC-singular point at the origin and $d= \dim \,{\rm aut} \,Q_{0}$. If $n >m$, then $d=2$, and if $n \leq m$, then $d=\infty$. Moreover, the estimate for the first case holds true also for all perturbations of $Q$ with terms of degree greater than $m+n$.