Geodesic
Classification of three-dimensional affine algebraic varieties that are quasi-homogeneous with respect to an algebraic group
Izvestiya. Mathematics , Tome 9 (1975) no. 3, pp. 535-576.

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In this article, we find all irreducible three-dimensional affine algebraic varieties that admit a quasi-transitive algebraic group of biregular automorphisms (that is, there is an orbit under the group action whose complement has dimension at most zero). The ground field is algebraically closed and has characteristic zero. Bibliography: 29 items. 
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V. L. Popov. Classification of three-dimensional affine algebraic varieties that are quasi-homogeneous with respect to an algebraic group. Izvestiya. Mathematics , Tome 9 (1975) no. 3, pp. 535-576. http://geodesic.mathdoc.fr/item/IM2_1975_9_3_a5/

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