  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
