Abstract
Through a reciprocal transformation induced by the conservation law , the Hunter–Saxton (HS) equation is shown to possess conserved densities involving arbitrary smooth functions, which have their roots in infinitesimal symmetries of , the counterpart of the HS equation under . Hierarchies of commuting symmetries of the HS equation are studied under appropriate changes of variables initiated by , and two of these are linearized while the other is identical to the hierarchy of commuting symmetries admitted by the potential modified Korteweg–de Vries equation. A fifth order symmetry of the HS equation is endowed with a sixth order hereditary recursion operator, which is proved to have a bi-Hamiltonian factorization, by its connection with the Fordy–Gibbons equation. These results reveal the origin for the rich and remarkable structures of the HS equation and partially answer the questions raised by Wang (2010 Nonlinearity 23 2009).
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