Exciton problem is solved in the two-dimensional Dirac model with allowance for strong electron-hole attraction. The exciton binding energy is assumed smaller than but comparable to the band gap. The exciton wavefunction is found in the momentum space as a superposition of all four two-particle states including electron and hole states with both positive and negative energies. The matrix element of exciton generation is shown to depend on the additional components of the exciton wavefunction. Both the Coulomb and the Rytova-Keldysh potentials are considered. The dependence of the binding energy on the coupling constant is analyzed for the ground and first excited exciton states. The binding energy and the oscillator strength are studied as functions of the environmental-dependent dielectric constant for real transition metal dichalcogenide monolayers. We demonstrate that the multicomponent nature of the exciton wavefunction is crucial for description of resonant optical properties of two-dimensional Dirac systems.
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