The F ( R , G ) theory of gravity, where R is the Ricci scalar and G is the Gauss-Bonnet invariant, is studied in the context of existence the Noether symmetries. The Noether symmetries of the point-like Lagrangian of F ( R , G ) gravity for the spatially flat Friedmann-Lemaitre-Robertson-Walker cosmological model is investigated. With the help of several explicit forms of the F ( R , G ) function it is shown how the construction of a cosmological solution is carried out via the classical Noether symmetry approach that includes a functional boundary term. After choosing the form of the F ( R , G ) function such as the case ( i ) : F ( R , G ) = f 0 R n + g 0 G m and the case ( i i ) : F ( R , G ) = f 0 R n G m , where n and m are real numbers, we explicitly compute the Noether symmetries in the vacuum and the non-vacuum cases if symmetries exist. The first integrals for the obtained Noether symmetries allow to find out exact solutions for the cosmological scale factor in the cases (i) and (ii). We find several new specific cosmological scale factors in the presence of the first integrals. It is shown that the existence of the Noether symmetries with a functional boundary term is a criterion to select some suitable forms of F ( R , G ) . In the non-vacuum case, we also obtain some extra Noether symmetries admitting the equation of state parameters w ≡ p / ρ such as w = − 1 , − 2 / 3 , 0 , 1 etc.
First integrals of holonomic systems without Noether symmetries
