  Let $K= \mathbf{Q}(\sqrt{d})$ be a quadratic field and $\mathcal{O}_{K}$ be
its ring of integers. We study the solvability of the Diophantine equation $r +
s + t = rst = 2$ in $\mathcal{O}_{K}$. We prove that except for $d= -7, -1, 17$
and $101$ this system is not solvable in the ring of integers of other
quadratic fields.
