  Let $X$ be a connected affine homogenous space of a linear algebraic group
$G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the
space of all algebraic vector fields on $X$ coincides with the Lie algebra
generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has
a $G$-invariant volume form $\omega$. We prove that the space of all
divergence-free (with respect to $\omega$) algebraic vector fields on $X$
coincides with the Lie algebra generated by divergence-free complete algebraic
vector fields on $X$ (including the cases when $X$ is a line or a torus). The
proof of these results requires new criteria for algebraic (volume) density
property based on so called module generating pairs.
