This paper deals with the singular semilinear elliptic system −Δu=μu|x|2+2αQ(x)(α+β)|x|s|u|α−2u|v|β+λ|u|q−2uin Ω,−Δv=μv|x|2+2βQ(x)(α+β)|x|s|u|α|v|β−2v+δ|v|q−2vinΩ,u=v=0on∂Ω, where Ω⊂RN(N≥3) is a smooth bounded domain, 0∈Ω and Ω is G-symmetric with respect to a subgroup G of O(N), 0≤μ<μ¯ with μ¯=(N−22)2, λ,δ≥0, 0≤s<2 and α,β>1 satisfy α+β=2∗(s)=2(N−s)N−2, 2<q<2∗=2NN−2, Q(x) is continuous and G-symmetric on Ω¯. Based upon the symmetric criticality principle of Palais and variational methods, we obtain several existence results of G-symmetric solutions under certain appropriate hypotheses on Q, q and the pair of parameters (λ,δ)∈R2.