Zero mean curvature surfaces in isotropic space with planar curvature lines

We give a comprehensive account of zero mean curvature surfaces with planar curvature lines in isotropic 3 -space. After providing a complete classification of all such surfaces, we show that they belong to a 1 -parameter family of surfaces. We then investigate their relationship to Thomsen-type surfaces in isotropic 3 -space, those zero mean curvature surfaces in isotropic 3 -space that are also affine minimal.