  We investigate how isospin affects the geometrical shape and energy of
classical soliton solutions of topological charges $B=1-4,8$ in the Skyrme
model. The novel approach in our work is that we study classically isospinning
Skyrmions beyond the rigid-body approximation; that is, we explicitly allow the
soliton solutions to deform and to break the symmetries of the static
configurations. Our fully three-dimensional relaxation calculations reveal that
the symmetries of isospinning Skyrme solitons can differ significantly from the
ones of the static configurations. In particular, isospinning Skyrmion
solutions can break up into lower-charge Skyrmions, can deform into new
solution types that do not exist at vanishing angular frequency $\omega$ or
energy degeneracy can be removed. These types of deformations have been largely
ignored in previous work on modeling nuclei by quantized Skyrmion solutions.
