  Lie group symmetry analysis for systems of coupled, nonlinear ordinary
differential equations is performed in order to obtain the entire solution
space to Einstein's field equations for vacuum Bianchi spacetime geometries.
The symmetries used are the automorphisms of the Lie algebra of the
corresponding three-dimensional isometry group acting on the hyper-surfaces of
simultaneity for each Bianchi Type, as well as the scaling and the time
reparametrization symmetry. A detailed application of the method is presented
for Bianchi Type IV. The result is the acquisition of the general solution of
Type IV in terms of sixth Painleve transcendent PVI, along with the known
pp-wave solution. For Bianchi Types I, II, V the known entire solution space is
attained and very briefly listed, along with two new Type V solutions of
Euclidean and neutral signature and a Type I pp-wave metric.
