  A spin-$\frac{1}{2}$ Kitaev sublattice interacting with a subsystem of
spinless fermions is studied on a honeycomb lattice when the fermion band is
half filled. The model Hamiltonian describes a topological Kondo lattice with
the Kitaev interaction, it is solved exactly by reduction to free Majorana
fermions in a static $\mathbb{Z}_2$ gauge field. An yet unsolved problem of a
hybridization of fermions and local moments in the Kondo lattice at low
temperatures is solved in the framework of the model proposed. The Kondo
hybridization gap is opened and the system is fixed in insulator and spin
insulator states, due to a spin-fermion nature of the gap. We will show that
the hybridization between local moments and itinerant fermions should be
understand as hybridization between corresponding Majorana fermions of the spin
and charge sectors. The RKKI interaction between local moments is not realized
in the model, a system demonstrates a `quasi Kondo' scenario of behavior with
realization chiral gapless edge states in topological nontrivial phases. The
ground-state phase diagram of the interacting subsystems calculated in the
parameter space is rich.
