Lie--Bäcklund symmetries of submaximal order of ordinary differential equations (Q1610757)

For a given \(n\)th-order ODE \(y^{(n)} = f(x,y,y',\dots, y^{(n-1)}\), canonical Lie-Bäcklund symmetries are given as the prolongations of generators \[ X=F(x,y_1, \dots,y_{p-1})\cdot {\partial\over \partial y},\quad p \leq n-1, \] where the higher derivatives are replaced using the differential equation, which is invariant under the corresponding action. Symmetries with \(p= n-2\) are called of submaximal order, the corresponding generators form then a linear space but not a Lie algebra in general. Using ideas of S. Lie, the authors prove that the number of symmetries of submaximal order is not greater than 3 for \(n=2\), not greater than 10 for \(n=3\) and not greater than 35 for \(n=4\) (the main result).