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We obtain an exact slowly rotating Einstein-bumblebee black hole solution bysolving the corresponding $rr$ and $t\phi$ components of the gravitationalfield 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 gravitationalfield equations and the bumblebee field motion equations by using thissolution. We find that in the case A, there exists a slowly rotating black holesolution indeed for arbitrary LV (Lorentz violation) coupling constant $\ell$;however as in the case B, there exists this slowly rotating solution if andonly if the coupling constant $\ell$ is as small as or smaller than the angularmomentum $a$. 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 inEinstein-bumblebee theory. It is similar as that in Einstein-aether theorywhere there exists only some slowly rotating black hole solutions. In order tostudy the effects of this Lorentz symmetry broken, we consider the black holegreybody 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.

Slowly rotating Einstein-bumblebee black hole solution and its greybody factor in a Lorentz violation model