Different definitions of homogeneity of real hypersurfaces in $\mathbb C^2$

The coincidence of two definitions of local homogeneity for real-analytic hypersurfaces in two-dimensional complex spaces is proved. It is shown that if any two germs of a Levi nondegenerate nonspherical surface $M$ are equivalent, then this surface has a local Lie group structure: $M$ then acts transitively on itself by left shifts, and each such shift is a local holomorphic transformation of $\mathbb C^2$.