Higher order first integrals of autonomous dynamical systems in terms of geometric symmetries

In general, a system of differential equations is integrable if there exist `sufficiently many' first integrals (FIs) so that its solution can be found by means of quadratures. Therefore, the determination of the FIs is an important issue in order to establish the integrability of a dynamical system. In this work, we consider holonomic autonomous dynamical systems defined by equations $\ddot{q}^{a}= -\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c} -Q^{a}(q)$ where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem which produces the FIs of any order of such systems in terms of the `symmetries' of the geometry defined by the quantities $\Gamma_{bc}^{a}(q)$. We apply the theorem to compute quadratic and cubic FIs of various dynamical systems.