Our construction method for numerical Godeaux surfaces relies mainly on two big steps: initially, the choice of a line in the complete quadratic intersection Q (defined by the entries of the matrix relPfaf) and, secondly, the choice of a solution of linear relations (defined by the entries of the matrix relLin). For a general line in Q, the solution space is a 4-dimensional linear space. However, if the the chosen line intersects special loci in Q, the dimension of the solution space may rise. All these possible exceptional loci are determined in this procedure.