Surfaces with planar lines of curvature and orthogonal systems of cycles

We determine the orthogonal systems of cycles (curves of constant geodesic curvature) on the hyperbolic plane, aiming at the classification of maximal surfaces with planar lines of curvature in the Minkowski space. The holomorphic functions representing normal maps of these surfaces are made explicit, in a one-to-one correspondence to the orthogonal systems of cycles. For geometrical comparison, we review the orthogonal systems of cycles on the round sphere and the classification of minimal surfaces with planar lines of curvature in the Euclidean space.