This paper deals with the existence of weak solutions to a class of degenerate and singular elliptic systems in ℝ N , N ≧2 of the form $$\left\{\begin{array}{l@{\quad}l}-\mathop{\mathrm{div}}(h_{1}(x)\nabla u)+a(x)u=f(x,u,v)&\mbox{in}\mathbb{R}^{N},\\-\mathop{\mathrm{div}}(h_{2}(x)\nabla v)+b(x)v=g(x,u,v)&\mbox{in}\mathbb{R}^{N},\end{array}\right.$$ where h i :ℝ N →[0,∞), h i ∈L loc 1 (ℝ N ), h i (i=1,2) are allowed to have “essential” zeroes at some points in ℝ N . Our proofs rely essentially on the critical point theory tools combined with a variant of the Caffarelli–Kohn–Nirenberg inequality in Nonlinear Differ. Equ. Appl. 7, 189–199, 2000.