  Many studies of synchronization properties of coupled oscillators, based on
the classical Kuramoto approach, focus on ensembles coupled via a mean field.
Here we introduce a setup of Kuramoto-type phase oscillators coupled via two
mean fields. We derive stability properties of the incoherent state and find
traveling wave solutions with different locking patterns; stability properties
of these waves are found numerically. Mostly nontrivial states appear when the
two fields compete, i.e. one tends to synchronize oscillators while the other
one desynchronizes them. Here we identify normal branches which bifurcate from
the incoherent state in a usual way, and anomalous branches, appearance of
which cannot be described as a bifurcation. Furthermore, hybrid branches
combining properties of both are described. In the situations where no stable
traveling wave exists, modulated quasiperiodic in time dynamics is observed.
Our results indicate that a competition between two coupling channels can lead
to a complex system behavior, providing a potential generalized framework for
understanding of complex phenomena in natural oscillatory systems.
