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Using a Lagrangian formalism, a three-parameter non-minimal Einstein-Maxwelltheory is established. The three parameters, $q_1$, $q_2$ and $q_3$,characterize the cross-terms in the Lagrangian, between the Maxwell field andterms linear in the Ricci scalar, Ricci tensor, and Riemann tensor,respectively. Static spherically symmetric equations are set up, and the threeparameters are interrelated and chosen so that effectively the system reducesto a one parameter only, $q$. Specific black hole and other type ofone-parameter solutions are studied. First, as a preparation, theReissner-Nordstr\"om solution, with $q_1=q_2=q_3=0$, is displayed. Then, weseek for solutions in which the electric field is regular everywhere as well asasymptotically Coulombian, and the metric potentials are regular at the centeras well as asymptotically flat. In this context, the one-parameter model with$q_1\equiv -q$, $q_2=2q$, $q_3=-q$, called the Gauss-Bonnet model, is analyzedin detail. The study is done through the solution of the Abel equation (the keyequation), and the dynamical system associated with the model. There is extrafocus on an exact solution of the model and its critical properties. Finally,an exactly integrable one-parameter model, with $q_1\equiv -q$, $q_2=q$,$q_3=0$, is considered also in detail. A special sub-model, in which theFibonacci number appears naturally, of this one-parameter model is shown, andthe corresponding exact solution is presented. Interestingly enough, it is asoliton of the theory, the Fibonacci soliton, without horizons and with a mildconical singularity at the center.

Non-minimal coupling for the gravitational and electromagnetic fields: black hole solutions and solitons