This is the "real-life" (but slightly more technical) version of a [question][1] I have asked recently. 

For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of all those lines in $\mathbb F_p^2$ parallel to the lines
  $$ X:=\{(x,0)\colon x\in\mathbb F_p \},
       \ Y:=\{(0,y)\colon y\in\mathbb F_p \},
       \ Z:=\{(z,z)\colon z\in\mathbb F_p \}, $$
respectively; thus, $|\mathcal L_X|=|\mathcal L_Y|=|\mathcal L_Z|=p$. Write
  $$ \chi(x,y) := \omega^x,\quad (x,y)\in\mathbb F_p^2, $$ 
where $\omega$ is a fixed primitive root of unity of degree $p$.
Given a set $S\subseteq\mathbb F_p^2$, with every element $s\in S$ associate a formal variable $x_s$, and consider the system of homogeneous linear equations
\begin{gather*}
 \sum_{s\in S\cap\ell} x_s = 0,\quad \ell\in\mathcal L_X\cup\mathcal L_Y, \\
 \sum_{s\in S\cap\ell} \chi(s)\,x_s=0, \quad \ell \in \mathcal L_Z;
\end{gather*}
notice that there are $3p$ equations and $|S|$ variables. Does there exist a set $S\subseteq\mathbb F_p^2$ of size $|S|<3p$ for which this system has a solution such that the set $\{s\in S\colon x_s\ne 0\}$ meets every line in $\mathbb F_p^2$?

[1]:https://mathoverflow.net/q/303359/9924