Symmetry of the problem in using the group of rotation of $\mathbb{R}^3$

Question

In Spectral Geometry in Non-standard Domains page $37$ point ($2$), it discusses of the symmetry of the problem in using the group of rotation of $\mathbb{R}^3$ with certain properties. Could anyone be able to explain to me why this point is important at the resolution of the spectrum on $\mathbb{S}^2$