  We study the properties of collective Yu-Shiba-Rusinov (YSR) states generated
by multiple magnetic adatoms (clusters) placed on the surface of a
superconductor. For magnetic clusters with equal distances between their
constituents, we demonstrate the formation of effectively spin-unpolarized YSR
states with subgap energies independent of the spin configuration of the
magnetic impurities. We solve the problem analytically for arbitrary spin
structure and analyze both spin-polarized (dispersive energy levels) and
spin-unpolarized (pinned energy levels) solutions. While the energies of the
spin-polarized solutions can be characterized solely by the net magnetic moment
of the cluster, the wave functions of the spin-unpolarized solutions
effectively decouple from it. This decoupling makes them stable against thermal
fluctuation and detectable in scanning tunneling microscopy experiments.
