I think I might try working with the second equation to get
[math]y = 1 + \frac{1}{2} * (uv - x^3z) \implies \\
\dfrac{\delta y}{\delta v} = \frac{1}{2}* \left ( u - x^3 * \dfrac{\delta z}{\delta v} - 3x^2z * \dfrac{\delta x}{\delta v} \right ) \text { and } \dfrac{\delta y}{\delta u} = \frac{1}{2} * \left ( v - x^3 * \dfrac{\delta z}{\delta u} - 3x^2z * \dfrac{\delta x}{\delta u} \right ).[/math]
Then, with a bunch of probably nasty algebra, you can eliminate y from the first and third equations. Now you have only two dependent variables. If you can eliminate either of them (for example z), you are down to a single equation relating two independent variable to one dependent variable.
There may of course be a much more elegant attack,