  We establish the existence of chimera states, simultaneously supporting
synchronous and asynchronous dynamics, in a network consisting of two
symmetrically linked star subnetworks consisting of identical oscillators with
shear and Kuramoto--Sakaguchi coupling. We show that the chimera states may be
metastable or asymptotically stable. If the intra-star coupling strength is of
order $\varepsilon$, the chimera states persist on time scales at least of
order $1/\varepsilon$ in general, and on time-scales at least of order
$1/\varepsilon^2$ if the intra-star coupling is of Kuramoto--Sakaguchi type. If
the intra-star coupling configuration is sparse, the chimeras are
asymptotically stable. The analysis relies on a combination of dimensional
reduction using a M\"obius symmetry group and techniques from averaging theory
and normal hyperbolicity.
