  The structure of the renormalization-group flows in a model with three
quartic coupling constants is studied within the $\epsilon$-expansion method up
to three-loop order. Twofold degeneracy of the eigenvalue exponents for the
three-dimensionally stable fixed point is observed and the possibility for
powers in $\sqrt{\epsilon}$ to appear in the series is investigated.
Reliability and effectiveness of the $\epsilon$-expansion method for the given
model is discussed.
