Some variational principles associated with ODEs of maximal symmetry. I: Equations in canonical form (Q1637400)

It is well-known that a linear ordinary differential equation (LODE) can always be reduced by a point transformation to the normal form \[ y^{(n)}+A_n^2y^{(n-2)}+\ldots+A_n^{n-1}y^{(1)}+A_n^ny=0. \] By exploiting the expression of the symmetry generators for LODE of the above form and of maximal symmetry, the present paper studies the existence for the corresponding class of LODE in canonical form \(y^{(n)}=0\). The variational and divergence symmetry algebras as well as first integrals are also discussed.