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We consider the relationship between symmetries of two-dimensional autonomousdynamical system in two common formulations; as a set of differential equationsfor the derivative of each state with respect to time, and a singledifferential equation in the phase plane representing the dynamics restrictedto the state space of the system. Both representations can be analysed withrespect to the symmetries of their governing differential equations, and weestablish the correspondence between the set of infinitesimal generators of therespective formulations. Our main result is to show that every generator of asymmetry of the autonomous system induces a well-defined vector fieldgenerating a symmetry in the phase plane and, conversely, that every symmetrygenerator in the phase plane can be lifted to a generator of a symmetry of theoriginal autonomous system, which is unique up to constant translations intime. The process of lifting requires the solution of a linear partialdifferential equation, which we refer to as the lifting condition. We discussin detail the solution of this equation in general, and exemplify the lift ofsymmetries in two commonly occurring examples; a mass conserved linear modeland a non-linear oscillator model.

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