Exact $(1+3+6)$-dimensional cosmological-type solutions in gravitational model with Yang-Mills field, Gauss-Bonnet term and $\mathrm{\Lambda }$-term

We consider $10$-dimensional gravitational model with $SO(6)$ Yang-Mills field, Gauss-Bonnet term and $\mathrm{\Lambda }$-term. We study so-called cosmological type solutions defined on product manifold $M=R\times R^3\times K$, where $K$ is $6d$ Calabi-Yau manifold. By putting the gauge field 1-form to be coinciding with 1-form spin connection on $K$, we obtain exact cosmological solutions with exponential dependence of scale factors (upon $t$-variable), governed by two non-coinciding Hubble-like parameters: $H>0$, $h$, obeying $H+2h\mathrm{\ne }0$. We also present static analogs of these cosmological solutions (for $H\mathrm{\ne }0$, $h\mathrm{\ne }H$ and $H+2h\mathrm{\ne }0$). The islands of stability for both classes of solutions are outlined.