Coefficients of prolongations for symmetries of ODEs (Q1777738)

The authors present a combinatorial approach to the calculation of the extensions of a Lie point symmetry of the form \(\xi(x,y)\partial_x+ \eta(x,y)\partial_y\) needed to calculate the determining equations and hence defy the symmetries of a given or the differential equation. As the first lemma proved is a result known in the literature since at least 1990, one is moved to view the rest of the paper with caution. Nevertheless, the combinatorial formulae do seem to work and one is inclined to accept the claim that there is a certain didactic value in the exercise for ordinary differential equations of moderate complexity. Unfortunately, the final example, \(y^{(6)}= y^2\), is somewhat at variance in this respect. Admittedly, the authors do not make a thorough investigation using the complete combinatorial arsenal assembled in the paper. They conclude that ax is the only symmetry of the equation although one can see by simple inspection that \(x\partial_x- 6y\partial_y\) is a second point symmetry. Naturally the equation does possess an infinite number of symmetries when one admits their full gamut. In the abstract, the authors refer to the use of MAPLE and MATHEMATICA `nowadays'. Maybe they are unaware of Alan Head's program based on MuMath from the late seventies and Clara Nucci's program based on REDUCE from the late eighties.