Essential constants for spatially homogeneous Ricci-flat manifolds of dimension 4+1

The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions-some already existing in the literature, and others believed to be new-are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization 'general' is given based on the counting of the essential constants, the line element of each model must contain; indeed, this is the basic contribution of the work. We give two different ways of calculating the number of essential constants for the simply transitive spatially homogeneous (4+1)-dimensional models. The first uses the initial value theorem; the second uses, through Peano's theorem, the so-called time-dependent automorphism inducing diffeomorphisms