**Abstract: ** Potential algebras can be used effectively in the analysis of the
quantum systems. We focus on the systems described by a separable, 2x2
matrix Hamiltonian of the first order in derivatives. We find integrals of
motion of the Hamiltonian that close centrally extended so(3), so(2,1) or
oscillator algebra. The algebraic framework is used in construction of
physically interesting solvable models described by (2+1) dimensional Dirac
equation. It is applied in description of open-cage fullerenes where the
energies and wave functions of low-energy charge-carriers are computed. The
potential algebras are also used in construction of shape-invariant,
one-dimensional Dirac operators. We show that shape-invariance of the
first-order operators is associated with the N=4 nonlinear supersymmetry
which is represented by both local and nonlocal supercharges. The relation
to the shape-invariant non-relativistic systems is discussed as well.