  We consider the relationship between symmetries of two-dimensional autonomous
dynamical system 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 the symmetries of their governing differential equations, and we
establish the correspondence between the set of infinitesimal generators of the
respective formulations. Our main result is to 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 autonomous system, which is unique up to constant translations in
time. The process of lifting requires the solution of a linear partial
differential equation, which we refer to as the lifting condition. We discuss
in detail the solution of this equation in general, and exemplify the lift of
symmetries in two commonly occurring examples; a mass conserved linear model
and a non-linear oscillator model.
