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Cosmological Solutions for the Geometrical Scalar-Tensor with the Potential Determined by the Noether Symmetry Approach

In this work we consider a scale-tensor theory in which the space-time is endowed with a Weyl integrable geometrical structure due to the Palatini variational method. Since the scalar field has a geometrical nature (related to non-metricity), the theory is known as \textit{Geometrical Scalar-Tensor}. On the framework of Weyl transformations, a non-minimally coupled scalar-tensor theory on the Jordan frame corresponds to a minimally coupled Einstein-Hilbert action on the Einstein frame. The scalar potential is selected by the Noether symmetry approach in order to obtain conserved quantities for the FRW cosmological model. Exact solutions are obtained and analyzed in the context of the cosmological scenarios consistent with an expanding universe. A particular case is matched in each frame and the role of scalar field as a dark energy component is discussed.