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The most general version of a renormalizable $d=4$ theory corresponding to adimensionless higher-derivative scalar field model in curved spacetime isexplored. The classical action of the theory contains $12$ independentfunctions, which are the generalized coupling constants of the theory. Wecalculate the one-loop beta functions and then consider the conditions forfiniteness. The set of exact solutions of power type is proven to consist ofprecisely three conformal and three nonconformal solutions, given by remarkablysimple (albeit nontrivial) functions that we obtain explicitly. The finitenessof the conformal theory indicates the absence of a conformal anomaly in thefinite sector. The stability of the finite solutions is investigated and thepossibility of renormalization group flows is discussed as well as severalphysical applications.Comment: LaTeX, 18 pages, no figure

A four-dimensional theory for quantum gravity with conformal and non-conformal explicit solutions