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We consider the highest anomalous dimension operator in the SU(2) sector ofplanar ${\cal N}=4$ SYM at all-loop, though neglecting wrapping contributions.In any case, the latter enter the loop expansion only after a preciselength-depending order. In the thermodynamic limit we write both a linearintegral equation for the Bethe root density and a linear system obeyed by thecommuting charges. Consequently, we determine the leading strong couplingcontribution to the density and from this an approximation to the leading andsub-leading terms of any charge $Q_r$: it scales as $\lambda ^{1/4-r/2}$, whichgeneralises the Gubser-Klebanov-Polyakov energy law. In the end, we brieflyextend these considerations to finite lengths and 'excited' operators by usingthe idea of a non-linear integral equation.

On the commuting charges for the highest dimension SU(2) operators in planar ${\cal N}=4$ SYM