  Epsilon-near-zero and epsilon near-pole materials enable reflective systems
supporting a class of symmetry-protected and accidental embedded eigenstates
(EE) characterized by a diverging phase-resonance. Here we show that pairs of
topologically protected scattering singularities necessarily emerge from EEs
when a non-Hermitian parameter is introduced, lifting the degeneracy between
oppositely charged singularities. The underlying topological charges are
characterized by an integer winding number and appear as phase vortices of the
complex reflection coefficient. By creating and annihilating them, we show that
these singularities obey charge conservation, and provide versatile control of
amplitude, phase and polarization in reflection, with potential applications
for polarization control and sensing.
