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Conservation laws and Exact Solutions in Brans-Dicke Cosmology with a Scalar Field

Scalar-tensor theories have drawn the attention of cosmologist for the past few years because they can provide mechanism to explain the observable phenomena. Moreover, the results of scalar-tensor theories can be applied in higher-order theories of gravity. In this work, we consider a Brans-Dicke scalar field cosmological model that includes a second scalar field minimally coupled to gravity. For the second scalar field we consider the two cases that there exist and there is not any interaction with the Brans-Dicke scalar field. For the two models of our consideration we prove the existence of scaling and de Sitter solutions. Furthermore, we perform an Obsiannikov's classification for the field equations, where we find all the point and contact transformations which leave the gravitational Action Integral invariant. Conservation laws are determined and we show how these can be applied to infer about the integrability of the cosmological models and find new analytic solutions.