Below we list some of them with a short explanation. We consider only gluon UGDs, thus we skip the subscripts in $\mathcal{F}_{g^{*}/A}$.

1.  pure BFKL equation\[enu:pure-BFKL-equation\]

    The equation in the leading logarithmic approximation reads [@Fadin:1975cb; @Kuraev:1977fs] $$\begin{gathered}
    \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\
    +\overline{\alpha}_{s}\int_{x}^{1}\frac{dz}{z}\,\int_{0}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right]\label{eq:BFKL}\end{gathered}$$ where $\overline{\alpha}_{s}=N_{c}\alpha_{s}/\pi$ with $N_{c}$ being the number of colors. The initial condition for the evolution is given by $\mathcal{F}_{0}$. The NLO BFKL equation is also known [@Fadin:1998py; @Ciafaloni:1998gs]. One of the drawbacks of the pure BFKL equation comes from the fact that $q_{T}^{2}$ of the gluons emitted along the ladder is unconstrained. Indeed, since in the BFKL regime the virtuality of the exchanged gluons is dominated by the transverse components, the resulting *kinematic constraint* reads [@Andersson:1995jt; @Kwiecinski:1996a] $$q_{T}^{2}<\frac{1-z}{z}\, k_{T}^{2}\approx\frac{1}{z}\, k_{T}^{2}.\label{eq:kinematic_constr}$$ This constraint is also often referred to as the consistency constraint.

2.  BFKL with the kinematic constraint (BFKL+C)

    To incorporate the consistency constraint one may include the appropriate step function into the real emission part of the BFKL. This operation, actually introduces some higher order corrections into the BFKL equation [@Kwiecinski:1996a]. In addition, one may introduce another class of sub-leading corrections by allowing the strong coupling constant to run with the local scale along the ladder. Finally, one may define the 