Search for numerical solutions of underdetermined systems of quadratic equations
This is a bit late for you, perhaps, but it might be worth looking at cylindrical algebraic decomposition. CAD takes a general statement in the real closed fields (i.e. an arbitrary boolean statement involving equalities and inequalities), and returns a finite set of "solutions", each of which is a "procedure" to find those solutions by assigning the unknown variables.

I haven't tried it on your example (because I don't have any software handy to do it!), but as an example, here is the CAD of $x^2 + y^2 = 1$:


$-1 &lt; x &lt; 1\,\wedge\,y = \sqrt{1-x^2}$
$-1 &lt; x &lt; 1\,\wedge\,y = -\sqrt{1-x^2}$
$x = -1\,\wedge\,y = 0$
$x = 1\,\wedge\,y = 0$


There are four "solutions" in this decomposition, and each one gives you a way to find it, by first determining an assignment to $x$, and using that to get an assignment to $y$.

This would work extremely well if you knew which variables were going to be constrained, because you could specify that they appear first in each "solution". In this case, if you added the constraint that $x = 0$, you would just go through each solution and find that $y=1$ and $y=-1$ are the only solutions, since the last two don't apply.