Introduction
Dynamical invariants were first used by Ermakov to show the connection between solutions of some special differential equations, referred to as Steen-Ermakov equations \cite{Ermakov_2008}. These equations were first studied by Steen \cite{Redheffer_2001} and then rediscovered by other authors \cite{Milne_1930,Pinney_1950}. After that, Ray and Reid used the Ermakov approach to construct invariants for a much broader class of differential equations \cite{Ray_1979,Ray_1979a,Ray_1980}. This purely mathematical interest was the start point of significant developments in classical and quantum dynamics.