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We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of
potential solenoid vector fields. We establish relations between axial-symmetric potential solenoid fields
and principal extensions of complex analytic functions into a special topological vector space containing
an infinite-dimensional commutative Banach algebra. In such a way we substantiate a method for explicit
constructing axial-symmetric potentials and Stokes flow functions by means of components of the
mentioned principal extensions and establish integral expressions for axial-symmetric potentials and
Stokes flow functions in an arbitrary simply connected domain symmetric with respect to an axis. The
obtained integral expression of Stokes flow function is applied for solving boundary problem about a
streamline of the ideal incompressible fluid along an axial-symmetric body. We obtain criteria of
solvability of the problem by means distributions of sources and dipoles on the axis of symmetry and
construct unknown solutions using multipoles together with dipoles distributed on the axis.