  We derive and discuss the complete set of exact solutions to the
one-dimensional Bogoliubov-de Gennes equations for small amplitude excitations
around general grey soliton solutions to the cubic nonlinear Schr\"odinger
equation. Our results extend the previously known case of the motionless dark
soliton background. We derive our non-zero frequency solutions using a variant
of the factorization method for Schr\"odinger equations with reflectionless
potentials. We also discuss the zero mode solutions at length.
