This paper continues the study of a rank factorization problem arising in gear fault surveillance. The structure of a class of solutions -- important in practice -- of the rank factorization problem is studied. We show that these solutions can be parametrized. Using module theory and computer algebra methods, the parameter space P is explicitly characterized and is shown to be the complementary of an algebraic set. Finally, a finite open cover of P is obtained and for each basic open subset of the cover of P, a closed-form solution is characterized.