  In this paper, we first construct a globally well-defined non-geometric
background which contains several branes in type II string theory compactified
on a 7-torus. One of these branes is called 5^2_2, which is a codimension-2
object and has a non-trivial monodromy given by a T-duality transformation. The
geometry near the 5^2_2-brane is shown to approach the non-geometric background
constructed in arXiv:1004.2521. We then construct the solution of a fundamental
string rotating along a non-trivial cycle in the 5^2_2 background. Although the
background is not axisymmetric in the usual sense, we show that it is actually
axisymmetric as a doubled geometry by explicitly finding a generalized Killing
vector. We perform a generalized coordinate transformation into a system where
the generalized isometry is manifest, and show that the winding and momentum
charges of the string solution is explicitly conserved in that system.
