A search for higher-dimensional arc planes

The concept of arc planes introduced byH. Groh (see [3]) has been extended to higher dimensions in [9]. By definition, a stable plane with point space ℝ2l is an arc plane if all vector space translations are automorphisms. The lines are either affine subspaces or graphs of non-affine functions. We examine these functions, especially their domains, and obtain information restricting the search for examples. Roughly speaking, one should start with a translation plane and substitute some lines with the graphs of suitable functions.