We consider a delay differential equation (DDE) model for mode-locked operation in class-A semiconductor lasers containing both gain and absorber sections. The material processes are adiabatically eliminated as these are considered fast in comparison to the delay time for a long cavity device. We determine the steady states and analyze their bifurcations using DDE-BIFTOOL [Engelborghs et al., ACM Trans. Math. Software 28, 1 (2002)]. Multiple forms of coexistence, transformation, and hysteretic behavior of stable steady states and fundamental periodic regimes are discussed in bifurcation diagrams.