A couple of catch-ups for the modified unit-circle zero location test

The unit-circle zero location test proposed by the author several decades ago has been improved to integrate nonessential singularities and received a version that remains faction-free for (real or Gaussian) integer polynomials with any singularity profile. In this paper we complement a modified form of the same test, proposed about a decade later, with a corresponding recursion that is not hampered by non-essential singularities and with a corresponding equally general integer preserving version.