  We present modified versions of existing criteria for the density property
and the volume density property of complex manifolds. We apply this methods to
show the (volume) density property for a family of manifolds given by
$x^2y=a(\bar z) + xb(\bar z)$ with $\bar z
=(z_0,\ldots,z_n)\in\mathbb{C}^{n+1}$ and volume form $\mathrm{d} x/x^2\wedge
\mathrm{d} z_0\wedge\ldots\wedge\mathrm{d} z_n$. The key step is showing that
in certain cases transitivity of the action of (volume preserving) holomorphic
automorphisms implies the (volume) density property, and then giving sufficient
conditions for the transitivity of this action. In particular, we show that the
Koras-Russell Cubic Threefold $\lbrace x^2y + x + z_0^2 + z_1^3=0\rbrace$ has
the density property and the volume density property.
