Existence and multiplicity result for the singular Toda system

We consider the Toda system on a compact surface (σ, g)-δu1=2ρ1(h1eu1∫σh1eu1dVg-1)-ρ2(h2eu2∫σh2eu2dVg-1)-4π∑j=1Jα1j(δpj-1),-δu2=2ρ2(h2eu2∫σh2eu2dVg-1)-ρ1(h1eu1∫σh1eu1dVg-1)-4π∑j=1Jα2j(δpj-1), where hiare smooth positive functions, ρiare positive real parameters, pjare given points on σ and αijare numbers greater than -1. We give existence and multiplicity results, using variational and Morse-theoretical methods. It is the first existence result when some of the αij's are allowed to be negative.