Abstract. Given an irreducible affine algebraic variety X of dimension n ≥ 2, we let SAut(X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x ∈ X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We provide also various modifications and applications.

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or $\C_+$. As another application of this approach we tackle the question (asked among others by F. Forstneri\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.Comment: to appear in Invent. Mat

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We study hypersurfaces of C n+2 x,u,v given by equations of form uv = p(x) where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.

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