A connection between Fano configurations and Minkowski planes of odd order (Q1120821)

Let \(M\) be a Minkowski plane and \(p{\bar \in}M\). Which projective planes are possible for being the projective closure \(\overline{M_p}\) of the affine derivation \(M_p\)? The closure of the circles of \(M\) not containing \(p\) are ovals through two special points of \(\overline{M_p}\). The author shows that for a plane of odd order those two points can not lie in a projective subplane \(\overline{M_p}\) of order two. This theorem can be applied to the Hughes plane of order 9 in which any pair of different points is contained in a subplane of order two.