Bound states in the continuum in periodic photonic systems like photonic crystal slabs are proved to be accompanied by vortex polarization singularities on the photonic bands in the momentum space. The winding structures of polarization states not only widen the field of topological physics but also show great potential that such systems could be applied in polarization manipulating. In this Letter, we report the phenomenon that by in-plane inversion (C2) symmetry breaking, pairs of circularly polarized states could spawn from the eliminated bound states in the continuum. Along with the appearance of the circularly polarized states as the two poles of the Poincaré sphere together with linearly polarized states covering the equator, full coverage on the Poincaré sphere could be realized. As an application, ellipticity modulation of linear polarization is demonstrated in the visible frequency range. This phenomenon provides a new degree of freedom in modulating polarization. The C points could also find applications in light-matter interactions. Further studying and manipulating the reported polarization singularities may lead to novel phenomena and physics in radiation modulating and topological photonics.