  We show that the one-dimensional projection of Chern-Simons gauged Nonlinear
Schrodinger model is equivalent to an Abelian gauge field theory of continuum
Heisenberg spin chain. In such a theory, the matter field has geometrical
meaning of coordinates in tangent plane to the spin phase space, while the U(1)
gauge symmetry relates to rotation in the plane. This allows us to construct
the infinite hierarchy of integrable gauge theories and corresponding magnetic
models. To each of them a U(1) invariant gauge fixing constraint of non-Abelian
BF theory is derived. The corresponding moving frames hierarchy is obtained and
the spectral parameter is interpreted as a constant-valued statistical gauge
potential constrained by the 1-cocycle condition.
