Geodesic
Quasihomogeneous affine algebraic varieties of the group~$SL(2)$
Izvestiya. Mathematics , Tome 7 (1973) no. 4, pp. 793-831.

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We classify up to $G$-isomorphism all normal affine irreducible quasihomogeneous (i.e. containing a dense orbit) varieties of the group $G=SL(2)$ which are defined over an algebraically closed field of characteristic zero. 
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V. L. Popov. Quasihomogeneous affine algebraic varieties of the group~$SL(2)$. Izvestiya. Mathematics , Tome 7 (1973) no. 4, pp. 793-831. http://geodesic.mathdoc.fr/item/IM2_1973_7_4_a3/

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