Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources

We consider the following Toda system \Delta u_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4\pi\gamma_{i}\delta_{0} $\text{in}$\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx < \infty, \forall 1\leq i \leq n, where ${\gamma }_i>-1$, ${\delta }_0$ is Dirac measure at 0, and the coefficients $a_{ij}$ form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: $$\sum _{j=1}^n{a}_{ij}{\int }_{{\mathbb{R}}^2}{e}^{u_j}dx=4\pi (2+{\gamma }_i+{\gamma }_{n+1-i}),\mathrm{\forall }1\le i\le n\mathrm{.}$$ This generalizes the classification result by Jost and Wang for ${\gamma }_i=0$, $\mathrm{\forall }1\le i\le n$. (ii) We prove that if ${\gamma }_i+{\gamma }_{i+1}+\mathrm{.}\mathrm{.}\mathrm{.}+{\gamma }_j\mathrm{\notin }\mathbb{Z}$ for all $1\leq i\leq j\leq n$,thenanysolution$u_i$ is $\mathit{\text{radially}}\mathit{\text{symmetric}}$ w.r.t. 0. (iii) We prove that the linearized equation at any solution is $\mathit{\text{non-degenerate}}$. These are fundamental results in order to understand the bubbling behavior of the Toda system.