  Unlike the well-known Mott's argument that extended and localized states
should not coexist at the same energy in a generic random potential, we provide
an example of a nearest-neighbor tight-binding disordered model which carries
both localized and extended states without forming the mobility edge (ME).
Unexpectedly, this example appears to be given by a well-studied
$\beta$-ensemble with independently distributed random diagonal potential and
inhomogeneous kinetic hopping terms. In order to analytically tackle the
problem, we locally map the above model to the 1D Anderson model with
matrix-size- and position-dependent hopping and confirm the coexistence of
localized and extended states, which is shown to be robust to the perturbations
of both potential and kinetic terms due to the separation of the above states
in space. In addition, the mapping shows that the extended states are
non-ergodic and allows to analytically estimate their fractal dimensions.
