  Time-dependent analytic solutions of the Einstein-Skyrme system --gravitating
Skyrmions--, with topological charge one are analyzed in detail. In particular,
the question of whether these Skyrmions reach a spherically symmetric
configuration for $t\rightarrow+\infty$ is discussed. It is shown that there is
a static, spherically symmetric solution described by the Ermakov-Pinney
system, which is fully integrable by algebraic methods. For $\Lambda>0$ this
spherically symmetric solution is found to be in a "neutral equilibrium" under
small deformations, in the sense that under a small squashing it would neither
blow up nor dissapear after a long time, but it would remain finite forever
(plastic deformation). Thus, in a sense, the coupling with Einstein gravity
spontaneously breaks the spherical symmetry of the solution. However, in spite
of the lack of isotropy, for $t \to\infty$ (and $\Lambda>0$) the space time is
locally flat and the anisotropy of the Skyrmion only reflects the squashing of
spacetime.
