$$\begin{aligned}
\label{masterint}
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} 
F_j (\omega, \eta) &=& 
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} 
[f_j(\omega) - f_j^{(0)}(\omega)] \\ \nonumber
& & -\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} \frac{1}{8\pi^2} b_j^{(\gamma)} 
\int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\ln(s/k_{\perp}^2)F_j (\omega, \eta' ) \\ \nonumber
& &+\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} c_j\int_{\mu^2}^{M^2} \frac{dk_{\perp}^2 }{k_{\perp}^2} 
\Big( 2f_j^{(0)}(\omega)F_j (\omega, \eta' ) + 
F_j^2(\omega, \eta') \Big) \Big] ~,\end{aligned}$$ where $\eta' = \ln(M^2/k_{\perp}^2)$. Differentiating Eq. (\[masterint\]) with respect to $\mu^2$ leads to the homogeneous partial differential equation for the on-shell amplitude $F_j (\omega, \eta)$: $$\label{master}
\frac{\partial F_j}{\partial \eta} = 
 - \frac{1}{8\pi^2} b_j^{(\gamma)} 
\Big( - \frac{\partial F_j}{\partial \omega} + \eta F_j\Big) + 
c_j \Big( 2 f_j^{(0)}(\omega) F_j + F_j^2 \Big)~,$$ where we have used the fact that $\ln(s/\mu^2)$, in Eq. (\[masterint\]), can be rewritten as $\ln(s/M^2) + \eta$ and that $\ln(s/M^2)$ corresponds to $- \partial/ \partial \omega$ .

Solutions to the evolution equations for collinear kinematics
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Let us consider first the particular case when $b_1^{(\gamma)} = 0$. It contributes to the forward leptonic, $e^+e^- \to \mu^+\mu^-$ annihilation and corresponds, in our notations, to the option $$\label{leptons}
 ~~Y_1 = Y_2 = -1.$$ Let us notice that $A_j$ with $j 