  We analyze earlier applications of perturbation theory by the moment method
(also called inner product method) to anharmonic oscillators. For concreteness
we focus on two-dimensional models with symmetry $C_{4v}$ and $C_{2v}$ and
reveal the reason why some of those earlier treatments proved unsuitable for
the calculation of the perturbation corrections for some excited states.
Point-group symmetry enables one to predict which states require special
treatment.
