  Using a Lagrangian formalism, a three-parameter non-minimal Einstein-Maxwell
theory is established. The three parameters, $q_1$, $q_2$ and $q_3$,
characterize the cross-terms in the Lagrangian, between the Maxwell field and
terms linear in the Ricci scalar, Ricci tensor, and Riemann tensor,
respectively. Static spherically symmetric equations are set up, and the three
parameters are interrelated and chosen so that effectively the system reduces
to a one parameter only, $q$. Specific black hole and other type of
one-parameter solutions are studied. First, as a preparation, the
Reissner-Nordstr\"om solution, with $q_1=q_2=q_3=0$, is displayed. Then, we
seek for solutions in which the electric field is regular everywhere as well as
asymptotically Coulombian, and the metric potentials are regular at the center
as 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 analyzed
in detail. The study is done through the solution of the Abel equation (the key
equation), and the dynamical system associated with the model. There is extra
focus 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 the
Fibonacci number appears naturally, of this one-parameter model is shown, and
the corresponding exact solution is presented. Interestingly enough, it is a
soliton of the theory, the Fibonacci soliton, without horizons and with a mild
conical singularity at the center.
