  The dynamics of the most general Bianchi IX cosmology with three time
dependent scale factors for the Einstein-Skyrme system is analyzed. For the
Skyrmion, a generalized hedgehog ansatz with unit baryon charge is introduced.
The most remarkable feature of this ansatz is that, in the above topologically
non-trivial sector with unit topological charge, the Skyrme field equations are
identically satisfied on any Bianchi IX metric. We will show that due to this
feature the complete set of coupled Einstein-Skyrme field equations can be
deduced from a suitable minisuperspace Lagrangian. The latter allows to perform
a systematic study of the integrability properties of the Einstein-Skyrme
system for the Bianchi IX cosmology. Moreover, some analytic and algebraic
solutions for the Einstein-Skyrme model are derived. Another remarkable
consequence of the present formalism is that it is possible to derive the
Wheeler de-Witt equation for the Bianchi IX metric in the Einstein-Skyrme
cosmology in which all the effects of the Skyrmion are encoded in an effective
potential of the minisuperspace Lagrangian.
