In this work we deal with non-discrete subgroups of ${\rm Diff}^{\omega} ({\mathbb S}^1)$, the group of orientation-preserving analytic diffeomorphisms of the circle. If $\Gamma$ is such a group, we consider its natural diagonal action ${\widetilde{\Gamma}}$ on the $n-$dimensional torus ${\mathbb T}^n$. It is then obtained a complete characterization of these groups $\Gamma$ whose corresponding ${\widetilde{\Gamma}}-$action on ${\mathbb T}^n$ is not piecewise ergodic (cf. Introduction) for all $n \in {\mathbb N}$ (cf. Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on ${\mathbb S}^1$ to general non-discrete subgroups of ${\mathbb S}^1$.