degenerate cusp: coordinate transformation

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I have a degenerate cusp singularity given by $xyz=x^2+y^3$. By an analysis of Shepherd-Barron, the standard form would be $z^2=y^2(y^n+x^2)$ for some $n>0$. I have a difficult time finding the transformation. Does anybody know how to do this? Thanks.

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Write the equation as $z(xy-z)=y^3$, then map $z$ to $-xy+z$, $x$ to $2x$ and $y$ to $-y$.