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For variational problems with $O(N)$-symmetry the existence of severalgeometrically distinct solutions had been shown by use of group theoreticapproach in previous articles. It was done by a crafty choice of a family $H_i\subset O(N)$ subgroups such that the fixed point subspaces $E^{H_i} \subset E$of the action in a corresponding functional space are linearly independent,next restricting the problem to each $E^{H_i}$ and using the Palais symmetryprinciple. In this work we give a thorough explanation of this approach showinga correspondence between the equivalence classes of such subgroups, partialorthogonal flags in $\mathbb{R}^N$, and unordered partitions of the number $N$.By showing that spaces of functions invariant with respect to different classesof groups are linearly independent we prove that the amount of series ofgeometrically distinct solutions obtained in this way grows exponentially in$N$, in contrast to logarithmic, and linear growths of earlier papers.

Geometrically distinct solutions given by symmetries of variational problems with the $O(N)$-symmetry