  The Kuramoto model describes the synchronization of coupled oscillators that
have different natural frequencies. Among the many generalizations of the
original model, Kuramoto and Sakaguchi (KS) proposed a {\it frustrated} version
that resulted in dynamic behavior of the order parameter, even when the average
natural frequency of the oscillators is zero. Here we consider a generalization
of the frustrated KS model that exhibits new transitions to synchronization.
The model is identical in form to the original Kuramoto model, but written in
terms of unit vectors. Replacing the coupling constant by a coupling matrix
breaks the rotational symmetry and forces the order parameter to point in the
direction of the eigenvector with highest eigenvalue, when the eigenvalues are
real. For complex eigenvalues the module of order parameter oscillates while it
rotates around the unit circle, creating active states. We derive the complete
phase diagram for Lorentzian distribution of frequencies using the Ott-Antonsen
ansatz. We also show that changing the average value of the natural frequencies
leads to further phase transitions where the module of the order parameter goes
from oscillatory to static.
