Let πl∞ be an affine translation plane of order qr with GF(q) in its kern. Suppose G is a subgroup of the translation complement of πl∞ which leaves invariant a set Δ of q + 1 slopes and acts transitively on l∞⧹Δ. We study the situation when G≌SL(n, q) or PSL(n, q).
We show that if G|Δ = identity, then πl∞ is a Hall plane, a Lorimer-Rahilly plane (LR-16) or a Johnson-Walker plane (JW-16). Moreover, if n⩾3, then G fixes Δ elementwise and πl∞ is LR-16 or JW-16.