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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., Comment: 13 pages, 1 figure. Accepted for publication in Symmetry

Multi-messenger observations of the transient sky to detect cosmic explosions and counterparts of gravitational wave mergers critically rely on orbiting wide-FoV telescopes to cover the wide range of wavelengths where atmospheric absorption and emission limit the use of ground facilities. Thanks to continuing technological improvements, miniaturised space instruments operating as distributed-aperture constellations are offering new capabilities for the study of high energy transients to complement ageing existing satellites. In this paper we characterise the performance of the upcoming joint SpIRIT + HERMES-TP/SP nano-satellite constellation for the localisation of high-energy transients through triangulation of signal arrival times. SpIRIT is an Australian technology and science demonstrator satellite designed to operate in a low-Earth Sun-synchronous Polar orbit that will augment the science operations for the equatorial HERMES-TP/SP. In this work we simulate the improvement to the localisation capabilities of the HERMES-TP/SP when SpIRIT is included in an orbital plane nearly perpendicular (inclination = 97.6$^\circ$) to the HERMES orbits. For the fraction of GRBs detected by three of the HERMES satellites plus SpIRIT, the combined constellation is capable of localising 60% of long GRBs to within ~ 30 deg$^2$ on the sky, and 60% of short GRBs within ~ 1850 deg$^2$. Based purely on statistical GRB localisation capabilities (i.e., excluding systematic uncertainties and sky coverage), these figures for long GRBs are comparable to those reported by the Fermi GBM. Further improvements by a factor of 2 (or 4) can be achieved by launching an additional 4 (or 6) SpIRIT-like satellites into a Polar orbit, which would both increase the fraction of sky covered by multiple satellite elements, and enable $\geq$ 60% of long GRBs to be localised within a radius of ~ 1.5$^\circ$ on the sky., Comment: 17 pages, 10 figures, 1 table. Accepted for publication in PASA