Degree counting for Toda system with simple singularity : one point blow up

In this paper, we study the degree counting formula of the rank two Toda system with simple singular source when ${\rho }_1\in (0,4\pi )\cup (4\pi ,8\pi )$ and ${\rho }_2\mathrm{\notin }4\pi \mathbb{N}\mathrm{.}$ The key step is to derive the degree formula of the shadow system, which arises from the bubbling solutions as ${\rho }_1$ tends to $4\pi$. In order to compute the topological degree of the shadow system, we need to find some suitable deformation. During this deformation, we shall deal with $\mathit{\text{new}}$ difficulty arising from the new phenomena: blow up does not necessarily imply concentration of mass. This phenomena occurs due to the collapsing of singularities. This is a continuation of the previous work Lee, Lin, Wei and Yang.