  In this paper, we study localization and mass spectrum of various matter
fields on a family of thick brane configurations in a pure geometric Weyl
integrable 5-dimensional space time, a non-Riemannian modification of
5-dimensional Kaluza--Klein (KK) theory. We present the shape of the
mass-independent potential of the corresponding Schr\"{o}dinger problem and
obtain the KK modes and mass spectrum, where a special coupling of spinors and
scalars is considered for fermions. It is shown that, for a class of brane
configurations, there exists a continuum gapless spectrum of KK modes with any
$m^2>0$ for scalars, vectors and ones of left chiral and right chiral fermions.
All of the corresponding massless modes are found to be normalizable on the
branes. However, for a special of brane configuration, the corresponding
effective Schr\"{o}dinger equations have modified P\"{o}schl-Teller potentials.
These potentials suggest that there exist mass gap and a series of continuous
spectrum starting at positive $m^2$. There are one bound state for spin one
vectors, which is just the normalizable vector zero mode, and two bound KK
modes for scalars. The total number of bound states for spin half fermions is
determined by the coupling constant $\eta$. In the case of no coupling
($\eta=0$), there are no any localized fermion KK modes including zero modes
for both left and right chiral fermions. For positive (negative) coupling
constant, the number of bound states of right chiral fermions is one less
(more) than that of left chiral fermions. In both cases ($\eta>0$ and
$\eta<0$), only one of the zero modes for left chiral fermions and right chiral
fermions is bound and normalizable.
