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Abstract: In this work, we analyze the effects of interactions that break rotational symmetry in a network of Kuramoto oscillators. The model can be seen as a generalization of the model for describing populations of pulse-coupled biological oscillators. We consider the case of a population of identical oscillators and apply the Watanabe-Strogatz approach to reduce the dimensionality of the original model. In our analytical analysis, we obtain their solutions, respective stabilities and a rich bifurcation landscape in the parameter space and compare them with the collective behavior of oscillators observed by extensive numerical simulations. We detect three main types of collective behavior, namely, synchronized steady state, a state not found in the original Kuramoto model; synchronized rotational state, a similar state recently observed in networks of Kuramoto oscillators with higher-order (triadic) interactions (and phase delay); and multicluster state, similarly manifested in the model with Hodgkin-Huxley neurons.