Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. Let $f:X\to M_{n}(\mathbb{C})$ be a continuous function. Under what of the following two conditions we can say: >There exist $x_{0}\in X$ such that the following integral is non invertible: $$ \int_{G} \rho(g)f(g.x_{0})d\mu $$ **Condition 1:** $G$ is abelian and $\rho$ is a non trivial representation. **Condition 2:** $\rho$ is an irreducible representation. This question is motivated by this [post](http://mathoverflow.net/questions/176607/integrals-of-representations-over-geodesics#comment458696_176607) and proposition $3$ of this [note](http://arxiv.org/abs/1110.0091).