  The electron states of gapped pseudospin-1 fermions of the $\alpha-{\cal
T}_3$ lattice in the Coulomb field of a charged impurity are studied. The free
$\alpha-{\cal T}_3$ model has three dispersive bands with two energy gaps
between them depending on the parameter $\Theta$ which controls the coupling of
atoms of honeycomb lattice with atoms in the center of each hexagon, thus,
interpolating between graphene $\Theta=0$ and the dice model $\Theta=\pi/4$.
The middle band becomes flat one with zero energy in the dice model. The bound
electron states are found in the two cases: the centrally symmetric potential
well and a regularized Coulomb potential of the charged impurity. As the charge
of impurity increases, bound state energy levels descend from the upper and
central continua and dive at certain critical charges into the central and
lower continuum, respectively. In the dice model, it is found that the flat
band survives in the presence of a potential well, however, is absent in the
case of the Coulomb potential. The analytical results are presented for the
energy levels near continuum boundaries in the potential well. For the genuine
Coulomb potential, we present the recursion relations that determine the
coefficients of the series expansion of wave functions of bound states. It is
shown that the condition for the termination of the series expansion gives two
equations relating energy and charge values. Hence, analytical solutions can
exist for a countably infinite set of values of impurity charge at fixed
$\Theta$.
