Classification of solutions to Toda systems of types $C$ and $B$ with singular sources

TL;DR

This paper extends the classification of solutions to Toda systems with singular sources from type A to types C and B, revealing their parametrization via Lie group subgroups and employing symmetry reductions and integrable system theories.

Contribution

It generalizes the classification of Toda systems with singular sources to types C and B, using symmetry reductions and Lie group decompositions.

Findings

Solution space parametrized by Lie group subgroups

Solutions characterized by minors of symplectic and orthogonal matrices

Method involves symmetry reductions from type A systems

Abstract

In this paper, the classification in [Lin,Wei,Ye] of solutions to Toda systems of type $A$ with singular sources is generalized to Toda systems of types $C$ and $B$. Like in the $A$ case, the solution space is shown to be parametrized by the abelian subgroup and a subgroup of the unipotent subgroup in the Iwasawa decomposition of the corresponding complex simple Lie group. The method is by studying the Toda systems of types $C$ and $B$ as reductions of Toda systems of type $A$ with symmetries. The theories of Toda systems as integrable systems as developed in [Leznov, Saveliev, Nie], in particular the $W$-symmetries and the iterated integral solutions, play essential roles in this work, together with certain characterizing properties of minors of symplectic and orthogonal matrices.