  We revisit the next-to-leading order~(NLO) correction to the eigenvalue of
the BFKL equation in the adjoint representation and investigate its properties
in analogy with the singlet BFKL in planar $\mathcal{N}=4$ super Yang-Mills
Theory~(SYM). We show that the adjoint NLO BFKL eigenvalue is needed to be
slightly modified in order to have a property of hermitian separability present
for the singlet BFKL. After this modification the adjoint NLO BFKL eigenvalue
is expressed through holomorphic and antiholomophic parts of the leading order
eigenvalue and their derivatives. The proposed choice of the modified NLO
expression is supported by the fact that it is possible to obtain the same
result in a relatively straightforward way directly from the singlet NLO BFKL
eigenvalue replacing alternating sums by non-alternating ones. This
transformation corresponds to changing cylindrical topology of the singlet BFKL
to the planar topology of the adjoint BFKL. We believe that the original NLO
calculation of Fadin and Lipatov is correct and valid for the computations of
the remainder function of the BDS amplitude. However, the notion of the adjoint
BFKL eigenvalue is vaguely defined due to removal of the infrared divergences
as well as redistributing NLO corrections between the kernel and impact
factors, and is to be modified to comply with properties of the singlet BFKL
equation. This, at first sight, a purely semantic difference may become
important in resolving the issue of the non-vanishing adjoint NNLO eigenvalue
in the limit of zero anomalous dimension $\nu$ and conformal spin $n$, which
contradicts the bootstrap condition of the BFKL equation.
