On one integrable cosmological model of the flat universe in k-essence

. The study of the origin and evolution of our Universe is one of the interesting and actual directions in modern physics and astrophysics. This paper considers the cosmological model of the Universe in the Einstein's theory of gravity and k -essence, where the gravitational field interacts in a non-minimal way with the scalar field φ . That is, in action, in the role of the matter field, we consider the special case of the Lagrange function for the essence. The corresponding field equations of the considered model are obtained. Also, particular solutions for the scale factor a ( t ) were obtained in the form of de Sitter's solution. Two solutions were found, for the potential energy V ( t ) and the scalar field φ ( t ), and their graphical solutions were also built. The analytical solutions obtained in this work are solutions of the considered integrable systems. These solutions are in good agreement with the available observational data and are able to describe the modern dynamics of the expansion of the Universe.