Higher-Order First Integrals of Autonomous Non-Riemannian Dynamical Systems

We consider autonomous holonomic dynamical systems defined by equations of the form $\ddot{q}^{a}=-\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$, where $\Gamma^{a}_{bc}(q)$ are coefficients a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ generalized forces. prove theorem which for these determines time-dependent first integrals (FIs) any order in systematic way, using `symmetries' geometry equations. demonstrate application to compute linear, quadratic, cubic FIs various Riemannian non-Riemannian systems.