  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.
