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Consider a system of $m$ balanced linear equations in $k$ variables withcoefficients in $\mathbb{F}_q$. If $k \geq 2m + 1$, then a routine applicationof the slice rank method shows that there are constants $\beta,\gamma \geq 1$with $\gamma < q$ such that, for every subset $S \subseteq \mathbb{F}_q^n$ ofsize at least $\beta \cdot \gamma^n$, the system has a solution$(x_1,\ldots,x_k) \in S^k$ with $x_1,\ldots,x_k$ not all equal. Building on aseries of papers by Mimura and Tokushige and on a paper by Sauermann, thispaper investigates the problem of finding a solution of higher non-degeneracy;that is, a solution where $x_1,\ldots,x_k$ are pairwise distinct, or even asolution where $x_1,\ldots,x_k$ do not satisfy any balanced linear equationthat is not a linear combination of the equations in the system.  In this paper, we focus on linear systems with repeated columns. For a largeclass of systems of this type, we prove that there are constants $\beta,\gamma\geq 1$ with $\gamma < q$ such that every subset $S \subseteq \mathbb{F}_q^n$of size at least $\beta \cdot \gamma^n$ contains a solution that isnon-degenerate (in one of the two senses described above). This class isdisjoint from the class covered by Sauermann's result, and captures the systemsstudied by Mimura and Tokushige into a single proof. Moreover, a special caseof our results shows that, if $S \subseteq \mathbb{F}_p^n$ is a subset suchthat $S - S$ does not contain a non-trivial $k$-term arithmetic progression(with $p$ prime and $3 \leq k \leq p$), then $S$ must have exponentially smalldensity.

On the size of subsets of $\mathbb{F}_q^n$ avoiding solutions to linear systems with repeated columns