  In this paper we construct and analyze new classes of wormhole and flux
tube-like solutions for the 5D vacuum Einstein equations. These 5D solutions
possess generic local anisotropy which gives rise to a gravitational running or
scaling of the Kaluza-Klein ``electric'' and ``magnetic'' charges of these
solutions. It is also shown that it is possible to self-consistently construct
these anisotropic solutions with various rotational 3D hypersurface geometries
(i.e. ellipsoidal, cylindrical, bipolar and toroidal). The local anisotropy of
these solutions is handled using the technique of anholonomic frames with their
associated nonlinear connection structures [vst]. Through the use of the
anholonomic frames the metrics are diagonalized, in contrast to holonomic
coordinate frames where the metrics would have off-diagonal components. In the
local isotropic limit these solutions are shown to be equivalent to spherically
symmetric 5D wormhole and flux tube solutions.
