  The Lieb lattice is one of the simplest lattices that exhibits both linear
Dirac-like and flat topological electronic bands. We propose to further tailor
its electronic properties through periodic 1D electrostatic superlattices
(SLs), which, in the long wavelength limit, were predicted to give rise to
novel transport signatures, such as the omnidirectional super-Klein tunnelling
(SKT). By numerically modelling the electronic structure at tight-binding
level, we uncover the evolution of the Lieb SL band structure from the discrete
all the way to the continuum regime and build a comprehensive picture of the
Lieb lattice under 1D potentials. This approach allows us to also take into
consideration the discrete lattice symmetry-breaking that occurs at the
well/barrier interfaces created by the 1D SL, whose consequences cannot be
explored using the previous low energy and long wavelength approaches. We find
novel features in the band structure, among which are intersections of
quadratic and flat bands, tilted Dirac cones, or series of additional
anisotropic Dirac cones at energies where the SKT is predicted. Such features
are relevant to experimental realizations of electronic transport in Lieb 1D SL
realized in artificial lattices or in real material systems like 2D covalent
organic/metal-organic frameworks and inorganic 2D solids.
