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We study properties of basic solutions in systems with dime delays and$S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) andrelative periodic solutions (MW solutions). It follows from the previous theorythat the number of CW solutions grows generically linearly with time delay$\tau$. Here we show, in particular, that the number of relative periodicsolutions grows generically as $\tau^2$ when delay increases. Thus, in suchsystems, the relative periodic solutions are more abundant than relativeequilibria. The results are directly applicable to, e.g., Lang-Kobayashi modelfor the lasers with delayed feedback. We also study stability properties of thesolutions for large delays.

Relative equilibria and relative periodic solutions in systems with time-delay and $S^{1}$ symmetry