  We obtain an exact slowly rotating Einstein-bumblebee black hole solution by
solving the corresponding $rr$ and $t\phi$ components of the gravitational
field equations in both cases: A, $b_\mu=(0,b(r),0,0)$; B,
$b_\mu=(0,b(r),\mathfrak{b}(\theta),0)$. Then we check the other gravitational
field equations and the bumblebee field motion equations by using this
solution. We find that in the case A, there exists a slowly rotating black hole
solution indeed for arbitrary LV (Lorentz violation) coupling constant $\ell$;
however as in the case B, there exists this slowly rotating solution if and
only if the coupling constant $\ell$ is as small as or smaller than the angular
momentum $a$. Till now there seems to be no full rotating black hole solution,
so one can't use the Newman-Janis algorithm to generate a rotating solution in
Einstein-bumblebee theory. It is similar as that in Einstein-aether theory
where there exists only some slowly rotating black hole solutions. In order to
study the effects of this Lorentz symmetry broken, we consider the black hole
greybody factor and find that when angular index $l=0$, the LV constant $\ell$
decreases the effective potential and enhances the absorption probability,
which is similar to that of the non-minimal derivative coupling theory.
