We consider the generation of thick brane configurations in a pure geometric Weyl integrable 5D spacetime 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 Poincaré 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 noncompact 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 nonsingular along the fifth dimension in the noncompact 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 Schrödinger’s equation form with a volcano potential of finite bottom in both the compact and the extended cases. We solve Schrödinger 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.

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