Density property for hypersurfaces $$UV = P({\bar X})

We study hypersurfaces of Cn+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.