By means of Riemann boundary value problems and of certain convenient systems of linear algebraic equations, this paper deals with the solvability of a class of singular integral equations with rotations and degenerate kernel within the case of a coefficient vanishing on the unit circle. All the possibilities about the index of the coefficients in the corresponding equations are considered and described in detail, and explicit formulas for their solutions are obtained. An example of application of the method is shown at the end of the last section.