Every one know solutions of the Diophatine equation $x^2+y^2=z^2$ which are given by formula $x=t(a^2-b^2)$, $y=t(2ab)$ and $z=t(a^2+b^2)$. In this exemple one proove that all the solutions are in this form.
My question is (Question 0): does all the solvable Diophantine equations have a unique complete description?
I think the word "description" is ambiguous. We could for exemple in a first approach require having description in polynomial form function of a finit number of parameters. So the more precise question could be:
Question 1: Is there a Diophantine equation (which has solution) for which we can proove that one parametrization in polynomial form is not enough to describe all solutions
A easier variant should be:
Question 2: Is there a Diophantine equation (which has solutions) for which we don't have yet find a unique polynomial expression for the solutions?