Noether Symmetry Analysis for Generalized Brans-Dicke Cosmology

We present the complete solution to the classification problem regarding the variational symmetries of the generalized Brans-Dicke cosmological model in the presence of a second scalar field minimally coupled to gravity and the generalized Brans-Dicke scalar field theory. Through the symmetry analysis, we were able to specify the functional form of the field equations such that they become integrable. Additionally, new families of integrable cosmological models are presented.