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We consider the 4+1 Einstein's field equations (EFE's) in vacuum, simplifiedby the assumption that there is a four-dimensional sub-manifold on which anisometry group of dimension four acts simply transitive. In particular weconsider the Abelian group Type 4A1; and thus the emerging homogeneoussub-space is flat. Through the use of coordinate transformations that preservethe sub-manifold's manifest homogeneity, a coordinate system is chosen in whichthe shift vector is zero. The resulting equations remain form invariant underthe action of the constant Automorphisms group. This group is used in order tosimplify the equations and obtain their complete solution space which consistsof seven families of solutions. Apart form the Kasner type all the othersolutions found are, to the best of our knowledge, new. Some of them correspondto cosmological solutions, others seem to depend on some spatial coordinate andthere are also pp-wave solutions.

The solution space to the Einstein's vacuum field equations for the case of five-dimensional Bianchi Type I (Type 4A1)