Generalised Symmetries and the Ermakov-Lewis Invariant

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number.Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry.For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous Ermakov-Lewis invariant, but in fact reveals a richer structure.