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

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.