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This manuscript explores the Darboux transformation employed in theconstruction of exactly solvable models for pseudospin-one particles describedby the Dirac-type equation. We focus on the settings where a flat band of zeroenergy is present in the spectrum of the initial system. Using the flat bandstate as one of the seed solutions substantially improves the applicability ofthe Darboux transformation, for it becomes necessary to ensure the Hermiticy ofthe new Hamiltonians. This is illustrated explicitly in four examples, where weshow that the new Hamiltonians can describe quasi-particles in Lieb latticewith inhomogeneous hopping amplitudes.

Reflectionless pseudospin-1 Dirac systems via Darboux transformation and flat band solutions