We address evolution of a spinor polariton condensate in radially periodic potentials.Such potentials allow for the observation of novel nonlinear excitations and support a variety of dynamically stable soliton states never demonstrated before in polariton condensates, including ring-like solitons with density peaks located in different radial minima of the potential and extended dynamically stable multiring patterns. Among the advantages of the system is that azimuthal modulational instabilities are suppressed due to dominating repulsive interactions between polaritons with the same spin, thereby allowing for the stabilization of radially-symmetric states. The representative feature of this system is that spin-orbit coupling between different spinor components requires them to carry different topological charges. Radially-symmetric states carrying different combinations of topological charges are discussed. Radially symmetric potentials also support stable rotating multipeaked solitons, whose properties unexpectedly depend not only on the magnitude of the rotation velocity, but also on its sign, i.e., on the rotation direction. The latter property is a consequence of spin-orbit coupling which breaks the equivalence between clockwise and counterclockwise rotations. The multiring structures are shown to be robust against unavoidable losses and are therefore amenable to observations with the presently available experimental techniques.Keywords polariton condensates, solitons, radial potentials, spin-orbit coupling Many-body interactions can play crucial role in quantum condensed matter systems and result in the appearance of the collective phenomena, which can not be qualitatively explained using the picture of non-interacting or weakly interacting particles. Well-known examples include such phenomena as superfluidity, superconductivity, ferromagnetism, fractional quantum Hall states, and many others. When strong inter-particle interactions are treated in the framework of the mean-field approximation, the behavior of the bosonic system can be well described by certain nonlinear wave equation for its macroscopic wavefunction, 1 while the particular form of this equation is defined by the physical nature of the system. The presence of nonlinearity drastically modifies the behavior of excitations in such systems and may result in spontaneous pattern formation, 2 turbulence, 3 appearance of self-sustained excitations, such as solitons 4 and topological vortex-carrying modes. 5The standard platform for the investigation of the coherent nonlinear phenomena in condensed matter is based on cold atoms. Such systems, however, require extremely low