$q_{T}^{2}$ integration region to lie away from the infrared nonperturbative region by separating the $\int_{0}^{k_{T0}^{2}}dq_{T}^{2}$ integration and moving it to the initial condition (the infrared cutoff $k_{T0}^{2}$ is taken to be of the order of $1\,\mathrm{GeV}$). The improved equation reads [@Kwiecinski:1997ee] $$\begin{gathered}
    \mathcal{F}\left(x,k_{T}^{2}\right)=\mathcal{F}_{0}\left(x,k_{T}^{2}\right)\\
    +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}dq_{T}^{2}\left[\frac{q_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)\Theta\left(k_{T}^{2}-zq_{T}^{2}\right)-k_{T}^{2}\mathcal{F}\left(\frac{x}{z},q_{T}^{2}\right)}{\left|q_{T}^{2}-k_{T}^{2}\right|}+\frac{k_{T}^{2}\mathcal{F}\left(\frac{x}{z},k_{T}^{2}\right)}{\sqrt{4q_{T}^{4}+k_{T}^{4}}}\right].\label{eq:BFKL_constr}\end{gathered}$$ Recently, it has been studied in the context of the Mueller-Navelet jets, that the energy-momentum conservation violation (which above is cured by a “brute force”) becomes less harmful when full NLO corrections are applied [@Ducloue:2014koa]. The effects of the kinematic constraints in the approximate form (\[eq:kinematic\_constr\]) as well as in the full form have been recently analyzed [@Deak:2015dpa] in the context of the CCFM evolution equation [Ciafaloni:1987ur,Catani:1989yc,Catani:1989sg,CCFMd]{}.

3.  BFKL with the kinematic constraint in re-summed form (BFKL+CR)

    The equation (\[eq:BFKL\_constr\]) can be casted in yet another form [@Kutak:2011fu] $$\begin{gathered}
    \mathcal{F}\left(x,k_{T}^{2}\right)=\tilde{\mathcal{F}}_{0}\left(x,k_{T}^{2}\right)\\
    +\overline{\alpha}_{s}\left(k_{T}^{2}\right)\int_{x}^{1}\frac{dz}{z}\,\int_{k_{T0}^{2}}^{\infty}\frac{d^{2}q_{T}}{\pi q_{T}^{2}}\Theta\left(q_{T}^{2}-\mu^{2}\right)\Delta_{R}\left(z,k_{T}^{2},\mu^{2}\right)\mathcal{F}\left(\frac{x}{z},\left|\vec{k}_{T}+\vec{q}_{T}\right|^{2}\right),\label{eq:BFKL+CR}\end{gathered}$$ where $$\Delta_{R}\left(z,k_{T}^{2},\mu^{2}\right)=\exp\left(-\overline{\alpha}_{s}\ln\frac{1}{z}\,\ln\frac{k_{T}^{2}}{\mu^{2}}\right)\label{eq:Regge_ff}$$ is the so-called Regge form factor. This form has been used in Ref. [@Kutak:2011fu] to propose a non-linear extension of the CCFM equation. The scale $\mu$ has been introduced to separate unresolved and resolved emissions in (\[eq:BFKL\_constr\]), i.e. the emissions with $q_{T}^{2}<\mu^{2}$ and $q_{T}^{2}>\mu^{2}$, and further the unresolved part was re-summed to obtain the Regge form factor. Note, that the UGDs undergoing 