Finite partially \(\{ 0,2\}\)-semiaffine linear spaces (Q1971024)

Partially \(\{ 0,2 \}\)-semiaffine linear spaces are linear spaces that admit a partition of the line set into two sets \({\mathcal V}\) and \({\mathcal I}\) such that: (I) for every line \(l \in {\mathcal V}\) and every point \(p\) not on \(l\), there are either 0 or 2 lines through \(p\) disjoint with \(l\), (II) through every point there are at most two lines of \({\mathcal I}\). Many examples of such linear spaces are given. A complete classification is given if different point degrees occur. If every point degree equals \(n+1\), then a classification is given if the number of points is at least \(n^2\).