Are there any solutions to the equation $s^{2}(1+t^{2})^{2}+t^{2}(1+s^{2})^{2}=u^2$ where $s,t,u\in \mathbb{Q}$ and $0 < s,t<1$? If so, is there a simple way to parametrize them all?

If I am understanding the geometry behind this problem, even if we pick a specific value for $t$ we are left with an elliptic curve, and it is possible there could be infinitely many solutions.

This question arises from some related (and somewhat esoteric) questions I have about rational points on unit circles. But this question looked "pretty" enough, I thought I'd ask the experts here.

Lectures on Elliptic Curves, or computer algebra systems will do it, to get the rational isomorphism. I suspect we can show that the Mordell-Weil group over $Q(t)$ is only $(Z/2)^2\oplus Z$ by descent or cohomology, but the details maybe tricky, especially if by hand. Another approach, is noting the rank 1 specializations and use Silverman's theorem, III.11 in hisAdvanced Topics in the Arithmetic of Elliptic Curves, effectively. Finally, Rankincreaseupon specialization is difficult, generally. – Junkie May 18 '11 at 2:07