This paper continues the study of a factorization problem arising in gear fault surveillance. The structure of a class of solutions -- interesting in practice -- of this factorization problem is studied. We show that these solutions can be parametrized. The parameter space ${\mathcal P}$ is proved to be the complementary of an algebraic set that is explicitly characterized based on module theory and computer algebra. A finite open cover of ${\mathcal P}$ is obtained and for each basic open subset of the cover, a closed-form solution is computed using computer algebra. Hence, the local structure of the solution space can be finely studied. Finally, we show that the existence of a single closed-form solution defined on the whole parameter space ${\mathcal P}$ is related to difficult problems in module theory.