We have a curvilinear orthogonal coordinate system defined with $u=xy$, $v=\frac{x^2 - y^2}{2}$, $z=z$.
First, calculate x and y. For them, I got $x=\sqrt{\frac{-2v\pm2\sqrt{v^2-u^2}}{2}}$ and $y=u/x$.
Then we have to calculate the Lamé coeffitiens, and after that the divergence.
I wonder if there is another method to define the x and y coordinates, or if I have miscalculated something; because with these expressions, I'm afraid I cannot carry out all the calculations.