  Spinor fields on surfaces of revolution conformally immersed into
3-dimensional space are considered in the framework of the spinor
representations of surfaces. It is shown that a linear problem (a 2-dimensional
Dirac equation) related with a modified Veselov- Novikov hierarchy in the case
of the surface of revolution reduces to a well-known Zakharov-Shabat system. In
the case of one-soliton solution an explicit form of the spinor fields is given
by means of linear Bargmann potentials and is expressed via the Jost functions
of the Zakharov-Shabat system. It is shown also that integrable deformations of
the spinor fields on the surface of revolution are defined by a modified
Korteweg-de Vries hierarchy.
