Symmetric vacuum scalar--tensor cosmology

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The existence of point symmetries in the cosmological field equations of generalized vacuum scalar--tensor theories is considered within the context of the spatially homogeneous cosmologies. It is found that such symmetries only occur in the Brans--Dicke theory when the dilaton field self--interacts. Moreover, the interaction potential of the dilaton must take the form of a cosmological constant. For the spatially flat, isotropic model, it is shown how this point symmetry may be employed to generate a discrete scale factor duality in the Brans--Dicke action.