  We analyze known results of next-to-next-to-leading(NNLO) singlet BFKL
eigenvalue in $N=4$ SYM written in terms of harmonic sums. The nested harmonic
sums building known NNLO BFKL eigenvalue for specific values of the conformal
spin have poles at negative integers. We sort the harmonic sums according to
the complexity with respect to their weight and depth and use their pole
decomposition in terms of the reflection identities to find the most
complicated terms of NNLO BFKL eigenvalue for an arbitrary value of the
conformal spin. The obtained result is compatible with the Bethe-Salpeter
approach to the BFKL evolution.
