The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization, and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The Sakaguchi-Kuramoto model is a generalization of the basic model that considers the presence of a phase-lag parameter in the interaction, thereby making it asymmetric between oscillator pairs. Here, we consider a further generalization, by adding an interaction that breaks the rotational symmetry of the model. The highlight of our study is the unveiling of a very rich phase diagram comprising both oscillatory and non-oscillatory synchronized phases as well as an incoherent phase: There are regions of two-phase as well as an interesting and hitherto unexplored three-phase coexistence arising from asymmetric interactions in our model.
