Given the simultaneous Diophantine equations,
$$u^2+v^2=w^2\tag{1}$$
$$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$
the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the elliptic curve,
$$x^4+y^4 = 193z^2$$
with initial solution $x,y = 18,31$. Or alternatively,
$$s^4+1 = 193t^2$$
which has initial rational point $s = 18/31$ (and an infinite more) and is birationally equivalent to an elliptic curve in Weierstrass form.
Question: Are there other co-prime $u,v$, such that (2) is solvable? (I did a quick search using Mathematica and didn't find any, though my search radius for $x,y$ could just have been too small.)