  We construct self-gravitating axially symmetric sphaleron solutions of the
3+1 dimensional Skyrme model coupled to Einstein gravity. The solutions are
static and asymptotically flat, they are characterized by two integers n and m,
where n is the winding numbers of the constituents and the second integer m
defines type of the solution. These configuration correspond to the chains of
charge n Skyrmions and charge -n anti-Skyrmions placed along the axis of
symmetry in alternating order. We investigate the dependency of the masses of
the gravitating sphalerons on the gravitational coupling. We find new chains of
self-gravitating |n| = 1 Skyrmions-anti-Skyrmions (S-A) which emerge at some
critical non-zero value of the gravitational coupling and do not have flat
space limit. In contrast, the branches of self-gravitating |n| $\ge$ 2 S-A
chains emerge from the corresponding flat space configurations. In both cases
these branches merge at some maximal value of the effective gravitational
coupling the branches of different type. The branch of gravitating S-A pair
extends all the way back to the limit of vanishing coupling constant where
solutions approach the corresponding generalised Bartnik-McKinnon solutions.
The upper branch of gravitating S-A-S chain exist up to some critical value of
the gravitational coupling at which the chain becomes broken. We further find
that for small values of the coupling constant on the upper branches, the
solutions correspond to composite systems, consisting of a scaled inner
Einstein-Yang-Mills solution and outer Skyrmions which are separating from the
inner configuration.
