  In this work we study the phase diagram of Kekul\'{e}-Kitaev model. The model
is defined on a honeycomb lattice with bond dependent anisotropic exchange
interactions making it exactly solvable in terms of Majorana representation of
spins in close analogy to the Kitaev model. However, the energy spectrum of
Majorana fermions has a multi-band structure characterized by Chern numbers 0,
$\pm$1, and $\pm2$. We obtained the phase diagram of the model in the plane of
exchange couplings and in the presence of a magnetic field and found chiral
topological and trivial spin-liquid ground states. In the absence of magnetic
field most part of the phase diagram is a trivial gapped phase continuously
connected to an Abelian phase, while in the presence of the magnetic field a
topological phase arises. Furthermore, motivated by recent thermal measurements
on the spin-liquid candidate $\alpha$-RuCl$_{3}$, we calculated the thermal
Hall conductivity at different regimes of parameters and temperatures and found
the latter is quantized over a wide range of temperatures.
