On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

We consider the relationship between symmetries of two-dimensional autonomous dynamical systems in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to their symmetries, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. We show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original system, which is unique up to constant translations in time. We exemplify the lift of symmetries in two cases; a mass conserved linear model and a non -linear oscillator.