ISL 2012 N1

Call admissible a set $A$ of integers that has the following property: $$\text { If } x, y \in A \text { (possibly } x=y \text { ) then } x^{2}+k x y+y^{2} \in A \text { for every integer } k \text {. }$$ Determine all pairs $m, n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.