Let $\mathbb{F}$ be a finite field of characteristic $2$. Let $L_m$ denote the set of lines in $\mathbb{F}^2$ with slope $m\in\mathbb{F}$, that is, all parallel lines of the form $y=mx+b$. Consider a subset $P$ of $n$ points in $\mathbb{F}^2$, then we call $L_m$ an even cover of $P$ if every line in $L_m$ contains an even number of points from $P$ (lines may contain no points).

Given an arbitrary set of $n$ points $P$, how many even covers are there of $P$?

We're looking for the sharpest possible bounds as a function of $n$. We've looked at small values of $n$, and so far there seems to be at most 2 even covers. If we change the requirement from containing an even number of points to containing at least 2 points, then we've found sets of points with more than two covers. Any ideas or references are very much welcome.