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

We consider the following Toda system, where γ _i>-1, δ 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: This generalizes the classification result by Jost and Wang for γ _i=0, ∀1≤i≤n. (ii) We prove that if γ _i+γ _i+1+⋯+γ _j∉ℤ for all 1≤i≤j≤n, then any solution u _i is radially symmetric w. r. t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.