  Variational and divergence symmetries are studied in this paper for the whole
class of linear and nonlinear equations of maximal symmetry, and the associated
first integrals are given in explicit form. All the main results obtained are
formulated as theorems or conjectures for equations of a general order. A
discussion of the existence of variational symmetries with respect to a
different Lagrangian, which turns out to be the most common and most readily
available one, is also carried out. This leads to significantly different
results when compared with the former case of the transformed Lagrangian. The
latter analysis also gives rise to more general results concerning the
variational symmetry algebra of any linear or nonlinear equations
