On Solutions for Singular Toda System on Riemann Surfaces with Boundary

This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\mathrm{\Sigma }$ with smooth boundaries $\mathrm{\partial }\mathrm{\Sigma }$. We establish the existence of solutions when the parameters are not critical, assuming that Euler characteristic $\chi (\mathrm{\Sigma })<1$ via analyzing the sublevels. Furthermore, we find a sufficient condition that ensures multiple solutions for generic potentials by Morse inequalities and a transversality theorem.