  We study the stationary Navier--Stokes equations in the region between two
rotating concentric cylinders. We first prove that, under the small Reynolds
number, if the fluid is axisymmetric and if its velocity is sufficiently small
in the $L^\infty$-norm, then it is necessarily a generalized Taylor-Couette
flow which is a new exact solution of the Navier--Stokes equations. If, in
addition, the associated pressure is bounded or periodic in the $z$-axis, then
it coincides with the well-known canonical Taylor-Couette flow. Next, we give a
certain bound of the Reynolds number and the $L^\infty$-norm of the velocity
such as the fluid is indeed, necessarily axisymmetric. It is clarified that
smallness of Reynolds number of the fluid in the two rotating concentric
cylinders governs both axisymmetry and the new exact form of the Taylor-Couette
flow.
