  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
$\rho_2\notin 4\pi\mathbb{N}.$ 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 \textit{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.
