Classification of Solutions to General Toda Systems with Singular Sources

We classify all the solutions to the elliptic Toda system associated to a general simple Lie algebra with singular sources at the origin and with finite energy. The solution space is shown to be parametrized by a subgroup of the corresponding complex Lie group. We also show the quantization result for the finite integrals. This work generalizes the previous works in [LWY12] and [Nie16] for Toda systems of types A and B, C. However, a more Lie-theoretic method is needed here for the general case, and the method relies heavily on the structure theories of the local solutions and of the W -invariants for the Toda system. This work will have applications to nonabelian Chern-Simons-Higgs gauge theory and to the mean field equations of Toda type.