  Anomalous dimension and higher conserved charges in the $sl(2)$ sector of
${\cal N}=4$ SYM for generic spin $s$ and twist $L$ are described by using a
novel kind of non-linear integral equation (NLIE). The latter can be derived
under typical situations of the SYM sectors, i.e. when the scattering need not
depend on the difference of the rapidities and these, in their turn, may also
lie on a bounded range. Here the non-linear (finite range) integral terms,
appearing in the NLIE and in the dimension formula, go to zero as $s\to
\infty$. Therefore they can be neglected at least up to the $O(s^0)$ order,
thus implying a linear integral equation (LIE) and a linear dimension/charge
formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case.
Importantly, these non-linear terms go faster than any inverse logarithm power
$(\ln s)^{-n}$, $n>0$, thus extending the linearity validity.
