Numerical modeling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behavior, and the fluid constitutive equation, which is an advection equation linked to the fluid history in differential models. In a former paper, the authors have proved that linear advection equations have only one solution in steady recirculating flows. This solution is found imposing the solution periodicity along the closed streamlines of the flow. The non-linear character of constitutive equations was taken into account in a second paper. In this case, standard strategies, as Picard or Newton techniques for example. were generalized to impose the solution periodicity for each linearized problem found in the iterative procedures. In the present paper, we treat other questions related to steady recirculating flows of non-Newtonian fluids. We will prove that in some cases the steady solution of the extra-stress tensor cannot be found as the steady state associated with the evolution problem. We will also prove that only one steady solution exists for multi-mode linear differential models. Finally, we will present some results concerning integral viscoelastic models in general steady recirculating flows.