  We use the Holder regularity analysis to study the symmetry breaking and
recovery due to a parametric potential generated via the strictly isospectral
factorization method. The initial potential is two-dimensional and periodic in
the two Cartesian directions, with the symmetry group $P_{4mm}$. The resulting
parametric isospectral potential display a P_m symmetry for values of the
parameter moderately close to the singular value gamma_s. However, at large
values of the parameter, visually around gamma=gamma_s+110, the original
symmetry is recovered. For a much higher precision value of the parameter for
this symmetry recovery, we show that the multifractal spectrum of the
parametric potential can be conveniently used. In the latter case, we obtain
gamma=gamma_s+201.085 for three decimal digits precision.
