  We construct analytic (3+1)-dimensional inhomogeneous and topologically
non-trivial pion systems using chiral perturbation theory. We discuss the
effect of isospin asymmetry with vanishing electromagnetic interactions as well
as some particular configurations with non-vanishing electromagnetic
interactions. The inhomogeneous configurations of the pion fields are
characterized by a non-vanishing topological charge that can be identified with
baryons surrounded by a cloud of pions. This system supports a topologically
protected persistent superflow. When the electromagnetic field is turned on the
superflow corresponds to an electromagnetic supercurrent.
