  Within the framework of the spontaneous Lorentz symmetry breaking, we
consider a metric--affine generalization of the gravitational sector of the
Standard--Model Extension (SME), including the Lorentz--violating (LV)
coefficients $u$ and $s^{\mu\nu}$. In this model, we derive the modified
Einstein field equations in order to obtain a new axisymmetric vacuum spinning
solution for a particular bumblebee's profile. Such a solution has the
remarkable property of incorporating the effects of Lorentz symmetry breaking
(LSB) through the LV dimensionless parameter $X=\xi b^2$, as the LSB is turned
off, $X=0$, we recover the well--established result, the Kerr solution, as
expected. Afterwards, we calculate the geodesics, the radial acceleration and
thermodynamic quantities for this new metric. We also estimate an upper bound
for $X$ by using astrophysical data of the advance Mercury's perihelion.
