  Let $X$ be an affine algebraic variety with a transitive action of the
algebraic automorphism group. Suppose that $X$ is equipped with several
non-degenerate fixed point free $SL_2$-actions satisfying some mild additional
assumption. Then we show 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.
