  We investigate the properties of the Kitaev-Kondo lattice model defined on a
bilayer honeycomb lattice by means of the SO(3) Majorana representation for
spin-$1/2$ moments. We first consider the pairing of neighboring sites for the
parent Kitaev spin liquid (KSL) Hamiltonian to render the Majorana and the
spin-$1/2$ Hilbert spaces perfectly equivalent to each other. As a consequence,
we demonstrate that this decoupling of the Kitaev interaction in terms of the
SO(3) Majorana fermions reproduces exactly the spectrum of the KSL model alone.
Then, by considering the effect of a local Kondo coupling $J_K$ in the model
and decoupling it in terms of an order parameter that physically must have a
finite staggering phase, we obtain that the system undergoes a quantum phase
transition from a fractionalized Fermi liquid to a nematic triplet
superconducting (SC) phase as $J_K$ is increased. Depending on the model
parameters, this SC phase can exhibit either Dirac points, Bogoliubov-Fermi
lines, or Bogoliubov-Fermi surfaces as nodal bulk manifolds. The surface states
in this latter case are also characterized by topologically protected
antichiral edge modes. The SC phase breaks time-reversal symmetry and exhibits
a coexistence of a dominant odd-frequency pairing with a small even-frequency
component for electronic excitations localized on sites of the same sublattice
of the system. Finally, we show that this SC phase is in fact a
pair-density-wave state, with Cooper pairs possessing a finite center-of-mass
momentum in zero magnetic field.
