  The notion of singular generalized Finsler spacetime and singular generalized
Berwald spacetime are introduced and their relevance for the description of
classical gravity discussed. A method to construct examples of such generalized
Berwald spacetimes is sketched. The method is applied at two different levels
of generality. First, a class of flat, singular generalized Berwald spacetimes
is obtained. Then in an attempt of further generalization, a class of non-flat
generalized Berwald spacetimes is presented and the associated Einstein field
equations are discussed. In this context, an argument in favour of a small
value of the cosmological constant is given. The physical significance of the
models is briefly discussed in the last section.
