On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb{Z}$ in ZFC

Following a recent paper by Herrlich and Tachtsis, we investigate in ZFC the following compactness question: for which unountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations is nontrivially solvable in $\mathbb{Z}$ provided that each its nonempty subsystem of cardinality $<\kappa$ is nontrivially solvable in $\mathbb{Z}$?