Poincar\'e-like approach to Landau theory. II. Simplifying the Landau-deGennes potential for nematic liquid crystals

In a previous paper we have discussed how the Landau potential (entering in Landau theory of phase transitions) can be simplified using the Poincar\'e normalization procedure. Here we apply this approach to the Landau-deGennes functional for the isotropic-nematic transitions, and transitions between different nematic phases, in liquid crystals. {We give special attention to applying our method in the region near the main transition point, showing in full detail how this can be done via a suitable simple modification of our Poincar\'e-like method. We also consider the question if biaxial phases can branch directly off the fully symmetric state; some partial results in this direction are presented.