  We show that the order of the cardinality of maximal complete $1$-systems of
loops on non-orientable surfaces is $\sim |\chi|^{2}$. In particular, we
determine the exact cardinality of maximal complete $1$-systems of loops on
punctured projective planes. To prove these results, we show that the
cardinality of maximal systems of arcs pairwise-intersecting at most once on a
non-orientable surface is $2|\chi|(|\chi|+1)$.
