Lie groupoid integration of singular isometries of the Poincaré disk

For every $n\ge 1$ there is a distinguished ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ action on the Poincaré disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal Möbius action, and for $n>1$ these vector fields have singularities at the origin and are incomplete, preventing integration to a global Lie group action. However, they naturally define an action Lie algebroid ${\mathcal{A}}_n={\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})⋉\text{Dbarstar}$ over the punctured disk. We construct an explicit Lie groupoid ${\mathcal{G}}_n$ integrating ${\mathcal{A}}_n$ and compare it to the \v Severa–Weinstein groupoid. Although ${\mathcal{G}}_n$ is not an action groupoid, its restriction to the boundary recovers an $n$-fold Möbius action on the boundary circle.