Lie–Bäcklund Symmetries of Submaximal Order of Ordinary Differential Equations

It is well known that the maximal order of Lie–Bäcklund symmetriesfor any nth-order ordinary differential equation is equal to n−1, and that the whole set of such symmetries forms aninfinite-dimensional Lie algebra. Symmetries of the order p≤n− 2 span a linear subspace (butnot a subalgebra) in this algebra. We call them symmetries of submaximalorder. The purpose of the article is to prove that for n≤ 4 this subspace is finite-dimensional and it's dimension cannot be greater than35 for n=4, 10 for n=3 and 3 for n=2. In the case n=3 this statementfollows immediately from Lie's result on contact symmetries ofthird-order ordinary differential equations. The maximal values of dimensions are reached, e.g., on the simplest equations y(n)=0.