Can Bertrand and Cournot oligopolies be combined?

There have been some recent attempts to combine Cournot and Bertrand duopolies in one single model. Unfortunately, these attempts do not work. A commodity cannot be homogenous and non-homogenous at the same time. It is always the consumers, who decide whether they perceive competing products as identical or as different brands for which they are willing to pay different prices. There is, of course, nothing that forbids the coexistence of both such consumer groups. Neither is there any obstacle for the competing sellers to sell to both markets. Then we only need an old idea from economic theory, i.e., price discrimination, to rectify the logic. By this the challenging combination idea comes on a stable footing. The model also results in some interesting mathematical facts, such as mulistability and coexistence of attractors.