It is shown that the magnetohydrodynamic (MHD) equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field are governed by a second-order partial differential equation for the poloidal magnetic flux function ψ coupled with a Bernoulli-type equation for the plasma density (which are identical in form to the corresponding ideal MHD equilibrium equations) along with the relation Δ*ψ = Vcσ (here Δ* is the Grad–Schlüter–Shafranov operator, σ is the conductivity and Vc is the constant toroidal-loop voltage divided by 2π). In particular, for incompressible flows, the above-mentioned partial differential equation becomes elliptic and decouples from the Bernoulli equation [H. Tasso and G. N. Throumoulopoulos, Phys. Plasma5, 2378 (1998)]. For a conductivity of the form σ = σ(R, ψ) (where R is the distance from the axis of symmetry), several classes of analytic equilibria with incompressible flows can be constructed having qualitatively plausible σ profiles, i.e. profiles with σ taking a maximum value close to the magnetic axis and a minimum value on the plasma surface. For σ = σ(ψ), consideration of the relation Δ*ψ = Vc σ(ψ) in the vicinity of the magnetic axis leads then to a proof of the non-existence of either compressible or incompressible equilibria. This result can be extended to the more general case of non-parallel flows lying within the magnetic surfaces.