Classification of complex simply connected homogeneous spaces of dimensions not greater than 2

We propose a classification of finite-dimensional complex Lie algebras of analytic vector fields on a complex plane and that of corresponding actions of Lie groups on complex two-dimensional manifolds. The mentioned algebras have been specified by S. Lie. More precisely, he has specified only vector fields, i.e., bases of the corresponding Lie algebras, rather than the structure of the algebras. No isomorphic algebras among the mentioned ones were specified. Therefore the Lie classification is far from complete; in this paper we complete it in one important case. We consider only a part of classification related to transitive actions of Lie groups.