  Two recently found coupled BPS submodels of the Skyrme model are further
analyzed. Firstly, we provide a geometrical formulation of the submodels in
terms of the eigenvalues of the strain tensor. Secondly, we study their
thermodynamical properties and show that the mean-field equations of state
coincide at high pressure and read $p=\bar{\rho}/3$. We also provide evidence
that matter described by the first BPS submodel has some similarity with a
Bose-Einstein condensate. Moreover, we show that extending the second submodel
to a non-BPS model by including certain additional terms of the full Skyrme
model does not spoil the respective ansatz, leading to an ordinary differential
equation for the profile of the Skymion, for any value of the topological
charge. This allows for an almost analytical description of the properties of
Skyrmions in this model. In particular, we analytically study the breaking and
restoration of the BPS property. Finally, we provide an explanation of the
success of the rational map ansatz.
