(Volume) Density Property of a family of complex manifolds including the Koras-Russell Cubic

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.