Existence of Lie–Bäcklund symmetries for 7th-order and higher equations in the k/(k+1) sequence

Ascertain whether the 7th-order and higher equations in the sequence u_t = (u_{(2k+1)x})^{−k/(k+1)} (for k ≥ 3) admit any Lie–Bäcklund symmetry generators beyond order 19, thereby determining whether those equations are symmetry-integrable.

Background

Within the first sequence u_t = (u_{(2k+1)x}){−k/(k+1)}, the authors verified that the 3rd- and 5th-order equations are symmetry-integrable. For the 7th-order member, they found no Lie–Bäcklund symmetries up to order 19.

They were unable to test beyond order 19 due to computational memory limitations and explicitly state that they cannot make any statement about the existence of Lie–Bäcklund symmetries for 7th-order and higher members of this sequence.

References

Due to the memory restrictions of our computer we are not able to consider Lie-Bäcklund symmetry generators of order higher than 19, so we can therefore not make any statement about the existence of Lie-Bäcklund symmetries for the equations in sequence ((i)) that are of order 7 or higher.

— Two sequences of fully-nonlinear evolution equations and their symmetry properties (2509.05535 - Euler et al., 5 Sep 2025) in Section 3