  This paper concerns the structural stability of smooth cylindrical symmetric
transonic flows in a concentric cylinder under helically symmetric perturbation
of suitable boundary conditions. The deformation-curl decomposition developed
by the second author and his collaborator is utilized to effectively decouple
the elliptic-hyperbolic mixed structure in the steady compressible Euler
equation. A key parameter in the helical symmetry is the step (denoted by
$\sigma$), which denotes the magnitude of the translation along the symmetry
axis after rotating one full turn. It is shown that the step determines the
type of the first order partial differential system satisfied by the radial and
vertical velocity. There exists a critical number $\sigma_{*}$ depending only
on the background transonic flows, such that if $0<\sigma<\sigma_{*}$, one can
prove the existence and uniqueness of smooth helically symmetric transonic
flows with nonzero vorticity.
