  We sketch the main features of the Noether Symmetry Approach, a method to
reduce and solve dynamics of physical systems by selecting Noether symmetries,
which correspond to conserved quantities. Specifically, we take into account
the vanishing Lie derivative condition for general canonical Lagrangians to
select symmetries. Furthermore, we extend the prescription to the first
prolongation of the Noether vector. It is possible to show that the latter
application provides a general constraint on the infinitesimal generator $\xi$,
related to the spacetime translations. This approach can be used for several
applications. In the second part of the work, we consider a gravity theory
including the coupling between a scalar field $\phi$ and the Gauss-Bonnet
topological term $\mathcal{G}$. In particular, we study a gravitational action
containing the function $F(\mathcal{G}, \phi)$ and select viable models by the
existence of symmetries. Finally, we evaluate the selected models in a
spatially-flat cosmological background and use symmetries to find exact
solutions.
