The Role of Three-dimensional Subalgebra in the Analysis of Hidden Symmetries of Differential Equations

Some new properties of symmetries that disappear as point symmetries after the first reduction of order of an ODE and reappear after the second are analyzed from the aspect of three-dimensional subalgebra of symmetries of differential equations. The form of a hidden symmetry is shown to consist of two parts, one of which always remains preserved as a point symmetry, and the second (fundamental) part which behaves as the complete hidden symmetry. Symmetry that disappears as point symmetry and remains hidden (non-local) during $n$ reductions of order before reappearing as a point symmetry is also introduced and termed convertible symmetry of order $n-1.$ We discuss the necessity for such classification in order to distinguish them from hidden symmetries of type I and type II, which are defined with respect to reduction of order by one.