Regularity and symmetry for solutions to a system of weighted integral equations

We study positive solutions of the following system of integral equations related to the weighted Hardy-Littlewood-Sobolev inequality:{u(x)=1|x|α∫Rnvq(y)|y|β|x−y|λdy,v(x)=1|x|β∫Rnup(y)|y|α|x−y|λdy, where u,v⩾0, 0