Geodesic
Quasihomogeneous affine surfaces
Izvestiya. Mathematics , Tome 5 (1971) no. 5, pp. 1057-1081

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We give a description of nonsingular affine surfaces in which one point is taken into another point by an automorphism of the surface, if these points do not belong to a finite set. 
@article{IM2_1971_5_5_a5,
     author = {M. Kh. Gizatullin},
     title = {Quasihomogeneous affine surfaces},
     journal = {Izvestiya. Mathematics },
     pages = {1057--1081},
     publisher = {mathdoc},
     volume = {5},
     number = {5},
     year = {1971},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1971_5_5_a5/}
}
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M. Kh. Gizatullin. Quasihomogeneous affine surfaces. Izvestiya. Mathematics , Tome 5 (1971) no. 5, pp. 1057-1081. http://geodesic.mathdoc.fr/item/IM2_1971_5_5_a5/