Abstract. Let X be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that X is equipped with several fixed point free non-degenerate SL 2 -actions satisfying some mild additional assumption. Then we prove that the Lie algebra generated by completely integrable algebraic vector fields on X coincides with the set of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form G/R where G is a linear algebraic group and R is its proper reductive subgroup.

A Danilov-Gizatullin surface is an affine surface V which is the complement of an ample section S of a Hirzebruch surface. The remarkable theorem of Danilov and Gizatullin states that the isomorphism class of V depends only on the self-intersection number S 2 . In this paper we apply their theorem to present V as the quotient of an affine threefold by a torus action, and to prove that the Lie algebra generated by the complete algebraic vector fields on V coincides with the set of all algebraic vector fields.

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We estabilish some connections among the Makar-Limanov invariant, the Derksen invariant, and the existence of flexible points on an affine variety.

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