  We study the extended Kitaev model called the K$\Gamma$ model, using a
perturbative expansion combined with a well-controlled mean-field approximation
and a cutting-edge exact diagonalization. In the phase diagram, we discover a
nematic Kitaev spin liquid and a Kekul\'e Kitaev spin liquid. The former
potentially explains the high-field nematic state with zero Chern number
experimentally observed in $\alpha$-RuCl$_3$, even for sufficiently small
values of $\Gamma/|K|$. The latter has a Majorana zero mode in its $Z_3$ vortex
core, which can be potentially controlled by the domain wall motion. This opens
a possible application of the quantum liquid crystal phases in the K$\Gamma$
model to the topological quantum computation.
