  A surface in the Teichm\"uller space, where the systole function attains its
maximum, is called a maximal surface. For genus two there exists a unique
maximal surface which is called the Bolza surface. In this article, we study
the complexity of the set of systolic geodesics on the Bolza surface. We show
that any non-systolic geodesic intersects the systolic geodesics in $2n$
points, where $n\geq 5$. Furthermore, we show that there are $12$ second
systolic geodesics on the Bolza surface and they form a triangulation of the
surface.
