  In this paper we study the dynamics of a vertically-emitting micro-cavity
operated in the Gires-Tournois regime that contains a semiconductor
quantum-well and that is subjected to strong time-delayed optical feedback and
detuned optical injection. Using a first principle time-delay model for the
optical response, we disclose sets of multistable dark and bright temporal
localized states coexisting on their respective bistable homogeneous
backgrounds. In the case of anti-resonant optical feedback, we disclose
square-waves with a periodic of twice the round-trip in the external cavity.
Finally, we perform a multiple time-scale analysis in the good cavity limit.
The resulting normal form is in good agreement with the original time-delayed
model.
