  We study actions of M\"obius group on two sub-populations in the solvable
chimera model proposed by Abrams et al. Dynamics of global variables are given
by two coupled Watanabe-Strogatz systems, one for each sub-population.
  At the first glance, asymptotic dynamics in the model seem to be very simple.
For instance, in the stable chimera state distributions of oscillators perform
a simple rotations after a certain (sufficiently large) moment. However, a
closer look unveils that dynamics are subtler that what can be observed from
evolution of densities of oscillators' phases. In order to gain the full
picture, one needs to investigate dynamics on the transformation group that
acts on these densities. Such an approach emphasizes impact of the "hidden"
variable that is not visible on macroscopic level.
