Consider a compact Lie group (the dimension of the group is greater than 1) and a representation $R^n$ of the group G. I am seeking an example of potential $p: R^n \rightarrow R$ which is invariant under the action of the group G such that $p’(v_0)=0$ and the isotropy group of $v_0$ is not trivial. If anyone knows this kind of potentials for instance in celestial mechanics, biology or chemistry. I know for example Newtonian potential of the N-body problem but isotropy groups of noncollinear central configurations are trivial.