Let $\mathcal L$ denote the set of all lines in $\mathbb F_p^2$ parallel to one of the lines
  $$ X:=\{(x,0)\colon x\in\mathbb F_p \},
       \ Y:=\{(0,y)\colon y\in\mathbb F_p \},
       \ Z:=\{(z,z)\colon z\in\mathbb F_p \}. $$
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
  $$ \sum_{s\in S\cap\ell} x_s = 0,\quad \ell\in\mathcal L; $$
thus, there are $|\mathcal L|=3p$ equations and $|S|$ variables. What is the smallest possible size of the set $S$ for which this system has a non-zero solution? Are there any sets of size $|S|<3p$ for which there are non-trivial solutions? If so, can these sets be characterized in a comprehensible way?