  The complete set of operators commuting with the Dirac Hamiltonian and exact
analytic solution of the Dirac equation for the two-dimensional Coulomb
potential is presented. Beyond the eigenvalue $\mu$ of the operator $j_{z}$,
two quantum numbers $\eta$ and $\kappa$ are introduced as eigenvalues of
hermitian operators $P=\beta\sigma_{z}'$ and $K=\beta(\sigma_{z}'l_{z}+1/2)$,
respectively. The classification of states according to the full set of
constants of motion without referring to the non-relativistic limit is
proposed. The linear Paschen-Back effect is analyzed using exact field-free
wave-functions as a zero-order approximation.
