  In this work, taking advantage of the Generalized Hedgehog Ansatz, we
construct new self-gravitating solitons in a cavity with mirror-like boundary
conditions for the SU(2) Non-linear Sigma Model and Skyrme model. For
spherically symmetric spacetimes, we are able to reduce the system to three
independent equations that are numerically integrated. There are two branches
of well-behaved solutions. The first branch is defined for arbitrary values of
the Skyrme coupling and therefore also leads to a gravitating soliton in the
Non-linear Sigma Model, while the second branch exists only for non-vanishing
Skyrme coupling. The solutions are quasi-static and in the first branch are
characterized by two integration constants that correspond to the frequency of
the phase of the Skyrme field and the value of the Skyrme profile at the
origin, while in the second branch the latter is the unique parameter
characterizing the solutions. These parameters determine the size of the
cavity, the redshift at the boundary of the cavity, the energy of the scalar
field and the charge associated to a U(1) global symmetry. We also show that
within this ansatz, assuming analyticity of the matter fields, there are no
spherically symmetric black hole solutions.
