  We substantiate the need for account of the proper electromagnetic field of
the electron in the canonical problem of hydrogen in relativistic quantum
mechanics. From mathematical viewpoint, the goal is equivalent to determination
of the spectrum of everywhere regular solutions to the self-consistent system
of Dirac and Maxwell equations (with external Coulomb potential). We
demonstrate that only particular classes of solutions, "nonlinear" analogues of
s- and p-states, can be obtained through decomposition of a solution in a
series, with respect to the fine structure constant parameter $\alpha$. In the
zero approximation at $\alpha \rightarrow 0$ the reduction to the
self-consistent non-relativistic system of Schr\"odinger-Poisson equations
takes place. For the latter, using both numerical and variational methods, we
obtain the solutions corresponding to the ground and set of excited states.
Spectrum of the binding energies with remarkable precision reproduces the
"Bohrian" dependence $W_n = W/ n^2$. For this, the ionization energy $W$ proves
to be universal yet about two times smaller than its observed value.
Possibility of the renormalization procedure and the problem of account for
relativistic corrections to the binding energies of order $\alpha^2$ are
considered
