Discrete orbits and special subgroups of $\text{Diff}$

The local topological dynamics of subgroups of ${\rm Diff} ({\mathbb C^n}, 0)$,withspecialemphasison${\rm Diff} ({\mathbb C^2}, 0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup of $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^2,0)$ possessing locally finite orbits is necessarily solvable. Other results and examples related to higher-dimensional generalizations of Mattei-Moussu’s celebrated topological characterization of integrability are also provided. These examples also settle a fundamental question raised by the previous work of Camara-Scardua.