Vanishing estimates for fully bubbling solutions of $SU(n+1)$ Toda Systems at a singular source

For Gauss curvature equation (or more general Toda systems) defined on two dimensional spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.