  We study the scale-invariant $O(g^4)$ kernel which appears as an infra-red
contribution in the BFKL evolution equation and is constructed via
multiparticle $t$-channel unitarity. We detail the variety of Ward identity
constraints and infra-red cancellations that characterize its infrared
behaviour. We give an analytic form for the full non-forward kernel. For the
forward kernel controlling parton evolution at small-x, we give an impact
parameter representation, derive the eigenvalue spectrum, and demonstrate a
holomorphic factorisation property related to conformal invariance. The results
show that, at next-to-leading-order, the transverse momentum infra-red region
may produce a strong reduction of the BFKL small-x behavior.
