I'm trying to decide whether the following set is a regular surface: $$\{(x^3 - 3xy^2, 3x^2y - y^3, 0) : (x, y) \in \mathbb{R}^2\}$$
I know that the map ${\bf x} : \mathbb{R^2} \to \mathbb{R^3}$ defined by ${\bf x}(x, y) = (x^3 - 3xy^2, 3x^2y - y^3, 0)$ doesn't work as a coordinate map, because the Jacobian determinant $\frac{\partial (x, y)}{\partial (x, y)}$ vanishes at $x = y = 0$. However I guess this doesn't rule out the possibility of other coordinate maps.
Obviously the surface isn't the graph of a function and I can't use the fact that it's a preimage of regular value of a function. So I'm thinking I have to find some coordinate maps which satisfy the appropriate properties, but I can't think what they would be. Can anyone help here? Is this set in fact not a regular surface, and if so how do I prove that?