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A filling of a closed hyperbolic surface is a set of simple
closed geodesics whose complement is a disjoint union of hyperbolic
polygons. The systolic length is the length of a shortest
essential closed geodesic on the surface. A geodesic is called systolic, if
the systolic length is realised by its length. For every 
$g\geq 2$, we construct closed hyperbolic surfaces of genus 
$g$ whose systolic geodesics fill the surfaces with
complements consisting of only two components. Finally, we remark that one
can deform the surfaces obtained to increase the systole.
