Painlevé III equation and Bianchi VII\(_0\) model (Q5929546)

Generally Bianchi models are spatially homogeneous spacetimes. Bianchi models can be considered as the homogeneous sector of general relativity. The reduced space of a cosmologial Bianchi \(\text{VII}_0\) model, with a signature \((+,+,-)\) is investigated. It can be shown that the dynamics of the sector of local degrees of freedom can be given by a Painlevé III differential equation. A zero-curvature representation is possible.