Basically I am looking for a linear system with many solutions from a finite set.

Choose a finite set of rationals $S$ and fix positive integer $k$.

Let $A$ be a linear system with $n$ variables $x_i$ and $n-k$ linearly independent equations.

Is the number of solutions of $A$ with $x_i \in S$ bounded by an absolute constant?

This might be related to integer programming or some variant of SAT.

If the number of solutions is bounded what is lower bound in terms if $k$ and $|S|$?