One of the greatest challenges in cosmology today is to determine the nature of dark energy, the sourse of the observed present acceleration of the Universe. Besides the vacuum energy, various dark energy models have been suggested. The Friedmann -Robertson -Walker (FRW) spacetime plays an important role in modern cosmology. In particular, the most popular models of dark energy work in the FRW spacetime. In this work, a new class of integrable FRW cosmological models is presented. These models induced by the well-known Painlevé equations. Some nonintegrable FRW models are also considered. These last models are constructed with the help of Pinney, Schrödinger and hypergeometric equations. Scalar field description and two-dimensional generalizations of some cosmological models are presented. Finally some integrable and nonintegrable F (R) and F (G) gravity models are constructed.

A step-by-step foundation for the differential character of the Universe's rotation is presented. First, invoking the concept of spacetime foam with spin, it is reasonable to assume that the very early Universe can be described by the Dirac equation. Second, it is shown using the Ehrenfest theorem that, from a classical point of view, the early Universe can be described by the Papapetrou equations. Third, it is stressed that our Universe can perform only rotational motion. It is shown based on the spin part of the Papapetrou equations that the Universe's rotation depends appreciably on the physical properties of a specific cosmological epoch. The rotational angular velocity is calculated for three basic cosmological epochs: the matter-dominated epoch, the transition period (from domination of matter to domination of vacuum), and the vacuum-dominated epoch.

A new version of the forming Universe large-scale structures is proposed, based on the refuse of analyses of only the gravitational instability of the cosmological substrate. Vacuum, i.e. the dominant nonbaryonic matter in the Universe, creates the antigravitational instability of the baryonic cosmic substrate itself and causes the formation of galaxies.

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