QUESTION Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
- $ x^2 + (x-u)^2\ =\ A^2$
- $ x^2 + (x+u)^2\ =\ B^2$
REMARK I have a family of pairs of quadruples $S\ T\subseteq\mathbb Z^2$, parametrized by $(u\ x)$, such that $S\ T$ have the same six distances but are not isometric (with respect to the Euclidean distance). All six distances of such quadruples are integers $\Leftrightarrow$ both integers $x^2+(x-u)^2$ and $x^2+(x+u)^2$ are full squares.