  The mobility edge (ME) is a critical energy delineates the boundary between
extended and localized states within the energy spectrum, and it plays a
crucial role in understanding the metal-insulator transition in disordered or
quasiperiodic systems. While there have been extensive studies on MEs in
one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the
investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In
the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with
imaginary quasiperiodic potentials applied solely to the vertices of the Lieb
lattice. By mapping this DLL model to the 2D NH Aubry-Andr{\'e}-Harper (AAH)
model, we analytically derive the exact ME and find it associated with the
absolute eigenenergies. We find that the eigenvalues of extended states are
purely imaginary when the quasiperiodic potential is strong enough.
Additionally, we demonstrate that the introduction of imaginary quasiperiodic
potentials does not disrupt the flat bands inherent in the system. Finally, we
propose a theoretical framework for realizing our model using the Lindblad
master equation. Our results pave the way for further investigation of exact NH
MEs and flat bands in 2D dissipative quasiperiodic systems.
