Further symmetry-integrable members of the k/(k+1) sequence

Determine whether there exist any additional symmetry-integrable equations in the sequence u_t = (u_{(2k+1)x})^{−k/(k+1)} beyond the known 3rd-order equation u_t = u_{3x}^{−1/2} and 5th-order equation u_t = u_{5x}^{−2/3}; either construct such equations by exhibiting infinitely many local Lie–Bäcklund symmetry generators or prove that no other members of the sequence are symmetry-integrable.

Background

The authors introduced two sequences of fully-nonlinear evolution equations and identified that in the first sequence, u_t = (u_{(2k+1)x})^{−k/(k+1)}, the 3rd- and 5th-order members are symmetry-integrable. Despite satisfying a necessary condition for symmetry-integrability, they could not find further symmetry-integrable examples in this sequence.

They explicitly declare an open problem to either find additional symmetry-integrable equations in the sequence or prove that these two are the only symmetry-integrable members.