Geodesic
Examples of nonhomogeneous quasihomogeneous surfaces
Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 43-60.

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Over a field of arbitrary positive characteristic we construct a nonsingular affine surface $X$ which is quasihomogeneous but not homogeneous. More precisely, we find generators of the group of automorphisms of this surface and show that there exists a point $\xi\in X$ which is invariant under all the automorphisms of $X$, while $\operatorname{Aut}(X)$ acts transitively on the points of $X-\xi$. 
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M. Kh. Gizatullin; V. I. Danilov. Examples of nonhomogeneous quasihomogeneous surfaces. Izvestiya. Mathematics , Tome 8 (1974) no. 1, pp. 43-60. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a3/

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[2] Gizatullin M. X., “Invarianty nepolnykh algebraicheskikh poverkhnostei, poluchaemye s pomoschyu popolnenii”, Izv. AN SSSR. Ser. matem., 35 (1971), 485–497 | MR | Zbl

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