  From rhythmic physiological processes to the collective behaviors of
technological and natural networks, coherent phases of interacting oscillators
are the foundation of the events' coordination leading a system to behave
cooperatively. We unveil the existence of a new of such states, occurring in
globally coupled nonidentical oscillators in the proximity of the point where
the transition from the system's incoherent to coherent phase converts from
explosive to continuous. In such a state, oscillators form quantized clusters,
where they are neither phase- nor frequency-locked. Oscillators' instantaneous
speeds are different within the clusters, but they form a characteristic cusped
pattern and, more importantly, they behave periodically in time so that their
average values are the same. Given its intrinsic specular nature with respect
to the recently introduced Chimera states, the phase is termed the {\it
Bellerophon} state. We provide analytical and numerical description of the
microscopic and macroscopic details of {\it Bellerophon} states, thus
furnishing practical hints on how to seek for the new phase in a variety of
experimental and natural systems.
