Geodesic
On actions of reductive groups with one-parameter family
Sbornik. Mathematics, Tome 188 (1997) no. 5, pp. 639-655

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Actions of reductive groups on normal algebraic varieties with one-parameter families of spherical orbits of maximal dimension are studied under the assumption that the categorical quotient for the action is one-dimensional. As an application, the classification of the actions of the group $SL_2$ on three-dimensional normal affine varieties is completed. The ground field $K$ is assumed to be algebraically closed and of characteristic zero. 
@article{SM_1997_188_5_a0,
     author = {I. V. Arzhantsev},
     title = {On actions of reductive groups with one-parameter family},
     journal = {Sbornik. Mathematics},
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     volume = {188},
     number = {5},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_5_a0/}
}
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I. V. Arzhantsev. On actions of reductive groups with one-parameter family. Sbornik. Mathematics, Tome 188 (1997) no. 5, pp. 639-655. http://geodesic.mathdoc.fr/item/SM_1997_188_5_a0/