  The mobility edge (ME) that marks the energy separating extended and
localized states is a central concept in understanding the metal-insulator
transition induced by disordered or quasiperiodic potentials. MEs have been
extensively studied in three dimensional disorder systems and one-dimensional
quasiperiodic systems. However, the studies of MEs in two dimensional (2D)
systems are rare. Here we propose a class of 2D vertex-decorated Lieb lattice
models with quasiperiodic potentials only acting on the vertices of a
(extended) Lieb lattice. By mapping these models to the 2D Aubry-Andr\'{e}
model, we obtain exact expressions of MEs and the localization lengths of
localized states, and further demonstrate that the flat bands remain unaffected
by the quasiperiodic potentials. Finally, we propose a highly feasible scheme
to experimentally realize our model in a quantum dot array. Our results open
the door to studying and realizing exact MEs and robust flat bands in 2D
systems.
