Lie groupoid integration of singular isometries of the Poincaré disk

For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ 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 $n>1$ the