  This manuscript explores the Darboux transformation employed in the
construction of exactly solvable models for pseudospin-one particles described
by the Dirac-type equation. We focus on the settings where a flat band of zero
energy is present in the spectrum of the initial system. Using the flat band
state as one of the seed solutions substantially improves the applicability of
the Darboux transformation, for it becomes necessary to ensure the Hermiticy of
the new Hamiltonians. This is illustrated explicitly in four examples, where we
show that the new Hamiltonians can describe quasi-particles in Lieb lattice
with inhomogeneous hopping amplitudes.
