Let $g,h,v,w$ be given rational functions. How to compute the map $(x,y) \mapsto (x', y')$ such that $(g(x,y), h(x,y))$ is sent to $(v(x,y), w(x,y))$? Thank you very much.
Edit: Sorry, my original question is confusing. My question is: How to compute the map $\eta: (x,y) \mapsto (p, q)$ ($p, q$ are rational functions in $x,y$) such that $(g(p,q), h(p,q))=(v(x,y), w(x,y))$?
For example, $g(x,y)=\frac{y^3}{x}$, $h(x,y)=y^3-x+2y$, $v(x,y)=y^3/x$, $w(x,y)=-h(x,y)$. The map we want is given by $(x,y)$ is sent to $(-x,-y)$.
This is not possible in general. For example, let $v(x,y)=x$, $w(x,y)=y$, and $g(x,y)=h(x,y)=x$. Then you are looking for $p,q$ such that $(g(p,q), h(p,q))=(x,y)$ which means $(p,p)=(x,y)$.