The effect on dynamo action of an anisotropic electrical conductivity conjugated to an anisotropic magnetic permeability is considered. Not only is the dynamo fully axisymmetric, but it requires only a simple differential rotation, which twice challenges the well-established dynamo theory. Stability analysis is conducted entirely analytically, leading to an explicit expression of the dynamo threshold. The results show a competition between the anisotropy of electrical conductivity and that of magnetic permeability, the dynamo effect becoming impossible if the two anisotropies are identical. For isotropic electrical conductivity, Cowling's neutral point argument does imply the absence of an azimuthal component of current density, but does not prevent the dynamo effect as long as the magnetic permeability is anisotropic.