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In this paper we perform a systematic study of vacuum spatially flat((3+D)+1)-dimensional Einstein-Gauss-Bonnet cosmological models. We considermodels which topologically are the product of two flat isotropic subspaces withdifferent scale factors. One of these subspaces is 3D and represents our spaceand the other is D-dimensional and represents extra dimensions. We consider noansatz of the scale factors, which makes our results quite general. With bothEinstein-Hilbert and Gauss-Bonnet contributions in play, the cases with D=1,D=2, D=3 and $D\geqslant 4$ have different dynamics due to different structureof the equations of motion. We analytically study equations of motion in allcases and describe all possible regimes. It appears that the only regimes withnonsingular future asymptotes are the Kasner regime in GR as well asexponential regimes. As of the past asymptotes, for a smooth transition onlyKasner regime in Gauss-Bonnet is an option. With that at hand, we are down onlyto two viable regimes -- "pure" Kasner regime (transition from high- tolow-energy Kasner regime) and a transition from high-energy Kasner toanisotropic exponential solution. It appears that these regimes take place fordifferent signs of the Gauss-Bonnet coupling $\alpha$: "pure" Kasner regimeoccur for $\alpha > 0$ at low D and $\alpha < 0$ for high D; anisotropicexponential regime is reached only for $\alpha > 0$. So if we restrainourselves with $\alpha > 0$ solutions, the only late-time regimes are Kasnerfor D=1, 2 and anisotropic exponential for $D\geqslant 2$. Also, low-energyKasner regimes ($a(t)\propto t^p$) have expansion rates for (3+1)-dimensionalsubspace ("our Universe") ranging from p=0.5 (D=1) to $p=1/\sqrt{3} \approx0.577$ ($D\to\infty$), which contradicts with dust-dominated Friedmannprediction (p=2/3).

Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions. Vacuum case