Four Symmetries of the KdV Equation

We revisit the symmetry structure of integrable PDEs, looking at specific example KdV equation. identify 4 nonlocal symmetries depending on a parameter, which we call generating symmetries. explain that since these are symmetries, their commutator algebra is not uniquely determined, and present three possibilities for algebra. In first version, 3 commute; this shows it possible to add further (nonlocal) commuting flows standard hierarchy. The second version consistent with Laurent expansions giving rise an infinite dimensional hidden KdV. third asymptotic large values KdV, hierarchy ``additional symmetries'', traditionally accepted (though also suffers from some ambiguity as additional nonlocal). how commute in can all be regarded infinitesimal double B\"acklund transformations. incorporate known equation, but exhibit remarkable novel structure, arising nonlocality. believe shared by other PDEs.