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We discuss a residual freedom of the next-to-leading BFKL eigenvalue thatoriginates from ambiguity in redistributing the next-to-leading~(NLO)corrections between the adjoint BFKL eigenvalue and eigenfunctions in planar$\mathcal{N}=4$ super-Yang-Mills~(SYM) Theory. In terms of the remainderfunction of the Bern-Dixon-Smirnov~(BDS) amplitude this freedom is translatedto reshuffling correction between the eigenvalue and the impact factors in themulti-Regge kinematics~(MRK) in the next-to-leading logarithmapproximation~(NLA). We show that the modified NLO BFKL eigenvalue suggested bythe authors can be introduced in the MRK expression for the remainder functionby shifting the anomalous dimension in the impact factor in such a way that thetwo and three loop remainder function is left unchanged to the NLA accuracy.

On a residual freedom of the next-to-leading BFKL eigenvalue in color adjoint representation in planar $\mathcal{N} = 4$ SYM