$ \bP^*_\mu $ are uniform. If $ G \neq {\left\{ {e}\right\}} $, then the solution $ \bP^*_\mu $ is non-strong.

This result says that, if $ G \neq {\left\{ {e}\right\}} $, then the case [[$ ({\bf C0}) $]{}]{} always fails. Theorem \[Intro exist\] will be restated in Theorem \[YorTh\]

We shall naturally define a group action of $ G $ on the set of solutions. Our key result is roughly stated as follows.

\[Intro action\] The action restricted on the set of extremal points is transitive.

This implies that the set of extremal points is exhausted by the $ G $-orbit of an arbitrary extremal point $ \bP^o $ and is homeomorphic to the homogeneous space $ G/H_\mu(\bP^o) $ where $ H_\mu(\bP^o) $ is the isotropic subgroup at the point $ \bP^o $. The precise statement of Theorem \[Intro action\] will be given in Theorem \[Gal1\].

Based on Theorem \[Intro action\] and fully employing the representation theory of compact groups, we generalize Theorem \[Intro YorTh\] to give a necessary and sufficient condition for the trichotomy [[$ ({\bf C1}) $]{}]{}-[[$ ({\bf C3}) $]{}]{} in terms of two subgroups $ H_\mu(\bP^o) $ and $ H_\mu^s $. Here $ H_\mu^s $ is a 