Systems of arcs on a torus with two punctures

For a compact surface $S$ with a finite set of marked points $P$, we define a 1-system to be a collection of arcs which are pairwise non-homotopic and intersect pairwise at most once. We prove that, up to equivalence, there are exactly 23 maximal 1-systems on $(S,P)$ when $S$ is a torus and $\mathrm{\mid }P\mathrm{\mid }=2$. Along the way, we generalize some of the results of a previous paper to the context of surfaces with boundary. In particular, we prove that the maximal cardinality of a 1-system on $(S,P)$ is $2\mathrm{\mid }\chi \mathrm{\mid }(\mathrm{\mid }\chi \mathrm{\mid }+1)-\frac{v}{2}$, where $\chi$ is the Euler characteristic of $(S,P)$ and $v$ is the number of marked points of $P$ in the boundary of $S$.