  The Kuramoto model serves as a paradigm to study the phenomenon of
spontaneous collective synchronization. We study here a nontrivial
generalization of the Kuramoto model by including an interaction that breaks
explicitly the rotational symmetry of the model. In an inertial frame (e.g.,
the laboratory frame), the Kuramoto model does not allow for a stationary
state, that is, a state with time-independent value of the so-called Kuramoto
(complex) synchronization order parameter $z\equiv re^{i\psi}$; Note that a
time-independent $z$ implies $r$ and $\psi$ both time independent, with the
latter fact corresponding to a state in which $\psi$ rotates at zero frequency
(no rotation). In this backdrop, we ask: Does the introduction of the
symmetry-breaking term suffice to allow for the existence of a stationary state
in the laboratory frame? Compared to the original model, we reveal a rather
rich phase diagram of the resulting model, with the existence of both
stationary and standing wave phases. While in the former the synchronization
order parameter $r$ has a long-time value that is time independent, one has in
the latter an oscillatory behavior of the order parameter as a function of time
that nevertheless yields a non-zero and time-independent time average. Our
results are based on numerical integration of the dynamical equations as well
as an exact analysis of the dynamics by invoking the so-called Ott-Antonsen
ansatz that allows to derive a reduced set of time-evolution equations for the
order parameter.
