  We investigate the properties of triply-periodic Skyrme crystals in the
generalized Skyrme model $\mathcal{L}_6 + \mathcal{L}_4 + \mathcal{L}_2+
\mathcal{L}_0$ with higher-derivative terms up to sixth order. Three different
symmetry breaking potential terms $\mathcal{L}_0$ are considered, the
generalized pion mass term, double vacuum potential and mixed potential.
Various scenario of phase transitions from the low density phase to the high
density phase are examined for different choices of the parameters of the
model. In particular, we investigated limiting behavior of the Skyrme crystals
in the truncated submodel without the Skyrme term $\mathcal{L}_4$ and/or
without the $\mathcal{L}_2$ term. We show that the Skyrme crystal may exist in
the pure $\mathcal{L}_4$ and $\mathcal{L}_6$ models and investigated the phase
structure of these solutions. Considering the near-BPS submodel, we found that
there are indications of the phase transition from a low density quasi-liquid
phase to the high density symmetric phase of the Skyrmionic matter.
