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I added an important result in the new version of my preprint: after the spinor field is eliminated from the Maxwell-Dirac electrodynamics, the resulting equations describe independent evolution of the electromagnetic field, in other words, a Cauchy problem can be formulated in terms of the electromagnetic field alone.

A new preprint:

We investigate whether the spinor field can be eliminated from the Maxwell-Dirac equations by differential algebraic methods. A generic truncated power series solution is constructed, the prolonged system of the Maxwell-Dirac electrodynamics is linearized about the solution, and the ranks of the associated coefficient matrices are computed. The results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. This strongly suggests that differential-algebraic elimination of the spinor field is possible.