I am looking for 2D transformation that lead to the following jacobian :
$$\frac{1}{1-xy}$$
I first tried with usual transformation (polar, afine, bipolar, elliptic and hyperbolic) but no success on it. After also trying some manual trials in wolframalpha I arrived to this best trial:
$$(x,y) \mapsto (\frac{y}{\sqrt{1-xy}}, -\frac{x}{\sqrt{1-xy}})$$
which leads to jacobian:
$$\frac{1}{(1-xy)^2}$$
I am looking for an example, despite a complete characterization (that means integrating partial differential system) could be also useful if provide further constraints in the possible solutions set (I guess there are infinite).