First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

Abstract

Complete sets of linearly independent first integrals are found for the most general form of linear equations of maximal symmetry algebra of order ranging from two to eight. The corresponding Hamiltonian systems are constructed and it is shown that their general solutions can also be found by a simple superposition formula from the solutions of a scalar second-order source equation.

Characterization of canonical classes for systems of linear ODEs

“…More recently, Krause and Michel [16] proved the converse of this result in the case of linear equations and showed that in fact a linear scalar equation is of maximal symmetry if and only if it is iterative. Some recent results have also been obtained about linear scalar equations of maximal symmetry, concerning their transformation properties [17,18], their coefficient characterization [19,20,17], their first integrals or Hamiltonian forms [21,22].…”