Integers of a quadratic field with prescribed sum and product

For given $k,\ell \in \mathbb Z$ we study the Diophantine system $$x+y+z=k, \quad x y z = \ell $$ for $x,y,z$ integers in a quadratic number field, which has a history in the literature. When $\ell =1$, we describe all such solutions; only for $k=5,6$, do there exist solutions in which none of $x,y,z$ are rational. The principal theorem of the paper is that there are only finitely many quadratic number fields $K$ where the system has solutions $x,y,z$ in the ring of integers of $K$. To illustrate the theorem, we solve the above Diophantine system for $(k,\ell )=(-5,7)$. Finally, in the case $\ell =k$, the system is solved completely in imaginary quadratic fields, and we give (conjecturally) all solutions when $\ell =k \leq 100$ for real quadratic fields.