  We naturally extend the theory of gravity with a conformally coupled scalar
field by only requiring conformal invariance of the scalar field equation of
motion and not of the action. The classically extended theory incorporates a
scalar-Gauss-Bonnet sector and has second-order equations of motion, belonging
to the Horndeski class. Remarkably, the theory features a purely geometrical
field equation that allows for closed-form black hole solutions and cosmologies
to be easily found. These solutions permit investigations of in-vogue
scalar-Gauss-Bonnet corrections to the gravitational action without the need of
resorting to approximations or numerical methods. We discuss on the connection
to the recently formulated 4D Einstein-Gauss-Bonnet theory of gravity.
