Geodesic
On $\operatorname{SL}_2$-actions of complexity one
Izvestiya. Mathematics , Tome 61 (1997) no. 4, pp. 685-698

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This paper is devoted to the actions of the group $\operatorname{SL}_2$ with two-dimensional generic orbit on normal three-dimensional affine algebraic varieties. We obtain a complete classification of such actions whenever the stabilizer of general position is the subgroup of monomial matrices. We also prove some results for arbitrary actions of complexity one of reductive groups. 
@article{IM2_1997_61_4_a0,
     author = {I. V. Arzhantsev},
     title = {On $\operatorname{SL}_2$-actions of complexity one},
     journal = {Izvestiya. Mathematics },
     pages = {685--698},
     publisher = {mathdoc},
     volume = {61},
     number = {4},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_4_a0/}
}
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I. V. Arzhantsev. On $\operatorname{SL}_2$-actions of complexity one. Izvestiya. Mathematics , Tome 61 (1997) no. 4, pp. 685-698. http://geodesic.mathdoc.fr/item/IM2_1997_61_4_a0/