  We study symmetries in the phase plane for separable, autonomous two-state
systems of ordinary differential equations (ODEs). We prove two main
theoretical results concerning the existence and non-triviality of two
orthogonal symmetries for such systems. In particular, we show that these
symmetries correspond to translations in the internal energy of the system, and
describe their action on solution trajectories in the phase plane. In addition,
we apply recent results establishing how phase plane symmetries can be extended
to incorporate temporal dynamics to these energy translation symmetries.
Subsequently, we apply our theoretical results to the analysis of three models
from the field of mathematical biology: a canonical biological oscillator
model, the Lotka--Volterra (LV) model describing predator-prey dynamics, and
the SIR model describing the spread of a disease in a population. We describe
the energy translation symmetries in detail, including their action on
biological observables of the models, derive analytic expressions for the
extensions to the time domain, and discuss their action on solution
trajectories.
