Multidimensional integrable vacuum cosmology with two curvatures

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$describingtheevolutionof$n$ Einstein spaces of non-zero curvatures is considered. For $n=2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1=$dim $ M_1, N_2 = $dim$ M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s\to +0$ is considered ($t_s$ is synchronous time). The exceptional (“Milne-type”) solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$\mathrm{.}Fordim$ M = 10, 11$and$3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using “curvature-splitting” trick. In the case $n=2$, $(N_1,N_2)=(6,3)$ a family of non-singular solutions with the topology $R^7\times M_2$ is found.