Locally affine geometries of order 2 where shrinkings are affine expansions

Given a locally affine geometry ⌈ of order 2 and a flag-transitive subgroup G ≤ Aut(⌈), suppose that the shrinkings of ⌈ are isomorphic to the affine expansion of the upper residue of a line of ⌈ by a homogeneous representation in a 2-group. We shall prove that, under certain hypotheses on the stabilizers Gp and Gl of a point p and a line l, we have G = RGp for a representation group R of Res(p). We also show how to apply this result in the classification of flag-transitive c-extended P-and T-geometries.