Translation structures with a principal line

Here we study affine parallel translation structures, both finite and infinite, with a principal line, that is a line which intersects every line not in its parallel class. These structures can be regarded also as (finite or infinite) translation transversal divisible designs. An algebraic characterization of these structures in terms of semidirect product of groups is provided and the main properties related to their group of automorphisms are inspected. The particular case of kinematic spaces is also taken into consideration.