LAUSR

In this paper we study a general 2×2 non-Abelian Chern–Simons–Higgs system of the form Δui+1e2(∑j=12Kjieuj−∑j=12∑k=12KkjKjieujeuk)=4π∑j=1Niδpij(x),i=1,2 over a flat 2-torus T2, where e>0, δp is the Dirac measure at p, Ni∈N (i=1,2), K is a non-degenerate 2×2 matrix of the form K=(1+a−a−b1+b), which may cover the physically interesting case when K is a Cartan matrix (of a rank 2 semisimple Lie algebra). Concerning the existence results of this type system over T2, usually in the literature there is a requirement that a,b>0. However, it is an open problem so far for the solvability about such system with a,b 0 such that this system admits a solution over the torus if 0