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 Σ with smooth boundaries ∂Σ. We establish the existence of solutions when the parameters are not critical, assuming that Euler characteristic χ(Σ)<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.