  The spectrum of a one-dimensional pseudospin-one Hamiltonian with a
three-component potential is studied for two configurations: (i) all the
potential components are constants over the whole coordinate space and (ii) the
profile of some components is of a rectangular form. In case (i), it is
illustrated how the structure of three (lower, middle and upper) bands depends
on the configuration of potential strengths including the appearance of flat
bands at some special values of these strengths. In case (ii), the set of two
equations for finding bound states is derived. The spectrum of bound-state
energies is shown to depend crucially on the configuration of potential
strengths. Each of these configurations is specified by a single strength
parameter $V$. The bound-state energies are calculated as functions of the
strength $V$ and a one-point approach is developed realizing correspondent
point interactions. For different potential configurations, the energy
dependence on the strength $V$ is described in detail, including its one-point
approximation. From a whole variety of bound-state spectra, four characteristic
types are singled out.
