Symmetries, integrals and solutions of ordinary differential equations of maximal symmetry

Second- and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the ‘exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.