Integration of $D$-dimensional cosmological models with two factor spaces by reduction to the generalized Emden–Fowler equation

The $D$-dimensional cosmological model on the manifold $M = R \times M_{1} \times \cdots\times M_{n}$, describing the evolution of Einstein factor spaces $M_i$ in the presence of a multicomponent perfect fluid source, is considered. The barotropic equation of state for the mass?energy densities and pressures of the components is assumed in each space. Where the number of non-Ricci-flat factor spaces and the number of perfect fluid components are both equal to two, the Einstein equations for the model are reduced to the generalized Emden–Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin using discrete-group analysis. We generate new integrable cosmological models using the integrable classes of this equation and present the corresponding metrics. The method is demonstrated for the special model with Ricci-flat spaces $M_1$ and $M_2$ and a two-component perfect fluid source