We say that a complex manifold $X$ has the density property if the set of the completely integrable holomorphic vector fields is dense in the space of all holomorphic vector fields of $X$. We first describe some remarkable properties of the group of holomorphic automorphism of a Stein manifold with the density property. Secondly, we present a theorem on the density property for homogeneous spaces endowed with "sufficiently many" $SL_2$-actions. In particular, the theorem applies to spaces of form $G/R$, where $G$ is a complex linear algebraic group and $R$ is a reductive subgroup.