  Cyclops states are intriguing cluster patterns observed in oscillator
networks, including neuronal ensembles. The concept of cyclops states formed by
two distinct, coherent clusters and a solitary oscillator was introduced in
[Munyayev {\it et al.}, Phys. Rev. Lett. 130, 107021 (2023)], where we explored
the surprising prevalence of such states in repulsive Kuramoto networks of
rotators with higher-mode harmonics in the coupling. This paper extends our
analysis to understand the mechanisms responsible for destroying the cyclops'
states and inducing new dynamical patterns called breathing and switching
cyclops' states. We first analytically study the existence and stability of
cyclops states in the Kuramoto-Sakaguchi networks of two-dimensional
oscillators with inertia as a function of the second coupling harmonic. We then
describe two bifurcation scenarios that give birth to breathing and switching
cyclops states. We demonstrate that these states and their hybrids are
prevalent across a wide coupling range and are robust against a relatively
large intrinsic frequency detuning. Beyond the Kuramoto networks, breathing and
switching cyclops states promise to strongly manifest in other physical and
biological networks, including coupled theta-neurons.
