The anisotropic EIT problem

It is now well known that an anisotropic conductivity is not uniquely determined by electrical data at the boundary. At least for the case of an analytic conductivity the only ambiguity is due to a possible distortion of the interior of the medium which would not change the boundary data. One needs then some additional data to determine the anisotropic conductivity uniquely from boundary data. If the conductivity is known to be isotropic then the unknown conductivity amounts to one unknown scalar field which is completely determined by the boundary data. One might expect that there are similar anisotropic problems where one unknown function can be determined from boundary data.