A surface in the Teichmuller 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 >= 5. For each non-negative integer n, we show the existence of curves on the Bolza surface which intersect the set of systolic geodesics at (10+6n) and (12+6n) points by construction. Furthermore, we show that there are exactly 12 second systolic geodesics on the Bolza surface and they form a triangulation of the surface.

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