In an earlier work we proposed a non-generational conjectural-derivation of all first class constraints (involving, only, variables compatible with canonical Poisson brackets) for “realistic” gauge (singular) field theories; and we verified the conjecture in cases of electromagnetic field, Yang-Mills fields interacting with scalar and spinor fields, and the gravitational field. Here, we will further verify our conjecture for the case of 't Hooft-Polyakov (HP) monopole's field (i.e. in the Higgs vacuum); and show that we will reproduce the results of Qandalji (Int. J. Theor. Phys. 45:1158, 2006), which were reached at using Dirac's standard multi-generational algorithm.