Classifying solutions of $\mathrm{S}\mathrm{U}(n+1)$ Toda system around a singular source

Consider a positive integer $n$ and ${\gamma }_1>-1,\cdots ,{\gamma }_n>-1$. Let $D=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$, and let $(a_{ij}{)}_{n\times n}$ denote the Cartan matrix of $\mathfrak{s}\mathfrak{u}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of $\mathrm{S}\mathrm{U}(n+1)$ Toda system, as discovered by Lin-Wei-Ye ({\it Invent Math}, {\bf 190}(1):169-207, 2012), we precisely characterize a solution $(u_1,\cdots ,u_n)$ to the $\mathrm{S}\mathrm{U}(n+1)$ Toda system \begin{equation*} \begin{cases} \frac{\partial^2 u_i}{\partial z\partial \bar z}+\sum_{j=1}^n a_{ij} e^{u_j}&=πγ_iδ_0,,{\rm on},, D\ \frac{\sqrt{-1}}{2},\int_{D\backslash {0}} e^{u_{i} }{\rm d}z\wedge {\rm d}\bar z &< \infty \end{cases} \quad $\text{for all}$\quad i=1,\cdots, n \end{equation*} using $(n+1)$ holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each $1\le i\le n$, $0$ represents the cone singularity with angle $2\pi (1+{\gamma }_i)$ for the metric $e^{u_i}\mathrm{\mid }\mathrm{d}z{\mathrm{\mid }}^2$ on $D\mathrm{\backslash }\{0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.