Dynamic games with multiple regimes frequently appear in economic literature. It is well known that, in regime-switching systems, novel effects emerge which do not appear in smooth problems. In this contribution, we explore in detail a differential game with regime switching and spillovers and study the behavior of optimal trajectories both in cooperative and non-cooperative differential game settings. We explicitly derive analytic solutions and point out cases where these solutions cannot be obtained via the application of maximum principle. We demonstrate that our approach that is based on an extension of the classical maximum principle agrees with the intuition given by the hybrid maximum principle while being more convenient for solving problems at hand.