  We suggest that the vacuum field equation in Finsler spacetime is equivalent
to vanishing of Ricci scalar. Schwarzschild metric can be deduced from a
solution of our field equation if the spacetime preserve spherical symmetry.
Supposing spacetime to preserve the symmetry of "Finslerian sphere", we find a
non-Riemannian exact solution of the Finslerian vacuum field equation. The
solution is similar to the Schwarzschild metric. It reduces to Schwarzschild
metric while the Finslerian parameter $\epsilon$ vanishes. It is proved that
the Finslerian covariant derivative of the geometrical part of the
gravitational field equation is conserved. The interior solution is also given.
We get solutions of geodesic equation in such a Schwarzschild-like spacetime,
and show that the geodesic equation returns to the counterpart in Newton's
gravity at weak field approximation. The celestial observations give constraint
on the Finslerian parameter $\epsilon<10^{-4}$. And the recent Michelson-Morley
experiment requires $\epsilon<10^{-16}$. The counterpart of Birkhoff's theorem
exist in Finslerian vacuum. It shows that the Finslerian gravitational field
with the symmetry of "Finslerian sphere" in vacuum must be static.
