  We construct the first analytic self-gravitating Skyrmions with higher Baryon
charge in four dimensions for the $SU(3)$-Skyrme-Einstein-$\Lambda$ theory by
combining the generalized hedgehog ansatz with the approach developed by
Balachandran et al. to describe the first (numerical) example of a non-embedded
solution. These are genuine $SU(3)$ analytic solutions instead of trivial
embeddings of $SU(2)$ into $SU(3)$ and its geometry is that of a Bianchi IX
Universe. The Skyrme ansatz is chosen in such a way that the Skyrme field
equations are identically satisfied in the sector with Baryon charge 4. The
field equations reduce to a dynamical system for the three Bianchi IX scale
factors. Particular solutions are explicitly analyzed. Traversable wormholes
with NUT-AdS asymptotics supported by a topologically non-trivial $SU(3)$-sigma
soliton are also constructed. The self-gravitating solutions admit also a
suitable flat limit giving rise to Skyrmions of charge 4 confined in a box of
finite volume maintaining the integrability of the $SU(3)$ Skyrme field
equations. This formalism discloses a novel transition at finite Baryon density
arising from the competition between embedded and non-embedded solutions in
which the non-embedded solutions prevail at high density while are suppressed
at low densities.
