  We consider the generation of thick brane configurations in a pure geometric
Weyl integrable 5D space time which constitutes a non-Riemannian generalization
of Kaluza-Klein (KK) theory. In this framework, we show how 4D gravity can be
localized on a scalar thick brane which does not necessarily respect reflection
symmetry, generalizing in this way several previous models based on the
Randall-Sundrum (RS) system and avoiding both, the restriction to orbifold
geometries and the introduction of the branes in the action by hand. We first
obtain a thick brane solution that preserves 4D Poincar'e invariance and breaks
Z_2-symmetry along the extra dimension which, indeed, can be either compact or
extended, and supplements brane solutions previously found by other authors. In
the non-compact case, this field configuration represents a thick brane with
positive energy density centered at y=c_2, whereas pairs of thick branes arise
in the compact case. Remarkably, the Weylian scalar curvature is non-singular
along the fifth dimension in the non-compact case, in contraposition to the RS
thin brane system. We also recast the wave equations of the transverse
traceless modes of the linear fluctuations of the classical background into a
Schr"odinger's equation form with a volcano potential of finite bottom in both
the compact and the extended cases. We solve Schr"odinger equation for the
massless zero mode m^2=0 and obtain a single bound wave function which
represents a stable 4D graviton. We also get a continuum gapless spectrum of KK
states with m^2>0 that are suppressed at y=c_2 and turn asymptotically into
plane waves.
