  In this work, we extend for the first time the spherically symmetric
Schwarzschild and Schwarzschild-De Sitter solutions with a Finsler-Randers-type
perturbation which is generated by a covector $A_\gamma$. This gives a locally
anisotropic character to the metric and induces a deviation from the Riemannian
models of gravity. A natural framework for this study is the Lorentz tangent
bundle of a spacetime manifold. We apply the generalized field equations to the
perturbed metric and derive the dynamics for the covector $A_\gamma$. Finally,
we find the timelike, spacelike and null paths on the Schwarzschild-Randers
spacetime, we solve the timelike ones numerically and we compare them with the
classic geodesics of general relativity. The obtained solutions are new and
they enrich the corresponding literature.
