Examples of nonhomogeneous quasihomogeneous surfaces

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 $X-\xi$.