  We generalize the spin Meissner effect for exciton-polariton condensate
confined in annular geometries to the case of non-trivial topology of the
condensate wavefunction. In contrast to the conventional spin Meissner state,
topological spin Meissner states can in principle be observed at arbitrary high
magnetic field not limited by the critical magnetic field value for the
condensate in a simply-connected geometry. One special example of the
topological Meissner states are half-vortices. We show that in the absence of
magnetic field half-vortices in a ring exist in a form of superposition of
elementary half-vortex states which resolves recent experimental results where
such puzzling superposition was observed. Furthermore, we show that if a pure
half-vortex state is to be observed, a non-zero magnetic field of a specific
magnitude needs to be applied. Studying exciton-polariton in a ring in presence
of TE-TM splitting, we observe spin Meissner states which break rotational
symmetry of the system by developing inhomogeneous density distributions. We
classify various states arising in presence of non-zero TE-TM splitting based
on what states they can be continued from by increasing the TE-TM splitting
parameter from zero. With further increasing TE-TM splitting, states with
broken symmetry may transform into stable half-dark solitons and therefore may
serve as a useful tool to generate various non-trivial states of a spinor
condensate.
