  A microscopic theory of electronic spectrum and superconductivity within the
$t$-$J$ model on the honeycomb lattice is formulated. The Dyson equation for
the normal and anomalous Green functions for the two-band model in terms of the
Hubbard operators is derived by applying the Mori-type projection technique.
The self-energy is evaluated in the self-consistent Born approximation for
electron scattering on spin and charge fluctuations induced by the kinematical
interaction for the Hubbard operators. Superconducting pairing mediated by the
antiferromagnetic exchange and spin fluctuations is discussed.
