  The time-delayed Adler equation is arguably the simplest model for an
injected semiconductor laser with coherent injection and optical feedback. It
is able to reproduce the existence of topological localized structures (LSs)
and their rich interactions. In this paper we perform the first extended
bifurcation analysis of this model and we explore the mechanisms by which LSs
emerge. We also derive the effective equations governing the motion of distant
LSs and we stress how the lack of parity in time-delayed systems leads to
exotic, non-reciprocal, interactions between topological localized states.
