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. 
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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/

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