Geodesic
On affine surfaces that can be completed by a nonsingular rational
Izvestiya. Mathematics , Tome 4 (1970) no. 4, pp. 787-810.

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It is proved that an affine nonsingular surface which can be completed by a nonsingular rational curve is quasi-homogeneous. The homogeneity of such surfaces is proved in the case of characteristic zero. 
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M. Kh. Gizatullin. On affine surfaces that can be completed by a nonsingular rational. Izvestiya. Mathematics , Tome 4 (1970) no. 4, pp. 787-810. http://geodesic.mathdoc.fr/item/IM2_1970_4_4_a5/

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