  The Kitaev model on the honeycomb lattice, while being integrable via the
spin-fermion mapping, has generally resisted an analytical treatment of the
far-from-equilibrium dynamics due to the extensive number of relevant
configurations of conserved charges. Here we study a close proxy of this model,
the isotropic Kitaev spin-$1/2$ model on the Bethe lattice. Instead of relying
on the spin-fermion mapping, we take a straightforward approach of solving
Heisenberg equations for a tailored subset of spin operators. The simplest
operator in this subset corresponds to the energy contribution of a single bond
direction. As an example, we calculate the time-dependent expectation value of
this observable for a factorized translation-invariant (or
staggered-translation-invariant) initial state with arbitrary initial
(staggered) polarization.
