Before revision:  We give a necessary and sufficient condition in terms of solutions to a system of monomial equations whose variables are in the binary octahedral group  .
After revision:  We give a necessary and sufficient condition in terms of solutions to a system of monomial equations whose variables are in the binary octahedral group when a boundary frame field and singularity graph have been fixed. This is phrased with respect to general cellular decompositions of the volume, which allows some flexibility in simplifying these systems .