  A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as
$n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least
as many monochromatic solutions to $L$ as a random 2-coloring. The notion of
common linear systems is analogous to that of common graphs, i.e., graphs whose
monochromatic density in 2-edge-coloring of cliques is asymptotically minimized
by the random coloring. Saad and Wolf initiated a systematic study on
identifying common linear systems, built upon the earlier work of
Cameron-Cilleruelo-Serra. When $L$ is a single equation, Fox-Pham-Zhao gave a
complete characterization of common linear equations. When $L$ consists of two
equations, Kam\v{c}ev-Liebenau-Morrison showed that irredundant $2\times 4$
linear systems are always uncommon. In this work, (1) we determine commonness
of all $2\times 5$ linear systems up to a small number of cases, and (2) we
show that all $2\times k$ linear systems with $k$ even and girth (minimum
number of nonzero coefficients of a nonzero equation spanned by the system)
$k-1$ are uncommon, answering a question of Kam\v{c}ev-Liebenau-Morrison.
