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In this paper we perform a systematic study of spatially flat[(3+D)+1]-dimensional Einstein-Gauss-Bonnet cosmological models with$\Lambda$-term. We consider models that topologically are the product of twoflat isotropic subspaces with different scale factors. One of these subspacesis three-dimensional and represents our space and the other is D-dimensionaland represents extra dimensions. We consider no {\it Ansatz} on the scalefactors, which makes our results quite general. With both Einstein-Hilbert andGauss-Bonnet contributions in play, the cases with $D=1$ and $D=2$ havedifferent dynamics due to the different structure of the equations of motion.We analytically study equations of motion in both cases and describe allpossible regimes. It is demonstrated that $D=1$ case does not have physicallyviable regimes while $D=2$ has smooth transition from high-energy Kasner toanisotropic exponential regime. This transition occurs for two ranges of$\alpha$ and $\Lambda$: $\alpha > 0$, $\Lambda > 0$ with $\alpha \Lambda\leqslant 1/2$ and $\alpha < 0$, $\Lambda > 0$ with $\alpha\Lambda < -3/2$. Forthe latter case if $\alpha\Lambda = -3/2$, extra dimensional part has $h\to 0$and so the size of extra dimensions (in the sense of the scale factor) isreaching constant value. We report substantial differences between $D=1$ and$D=2$ cases and between these cases and their vacuum counterparts, describefeatures of the cases under study and discuss the origin of the differences.

Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Low-dimensional $\Lambda$-term case