In Postel-Vinay & Robin 2002 they show an equation: $$\left\{\delta+\mu+\lambda_{1} \bar{F}(p)\right\} \ell(\varepsilon, p)=\left\{(\delta+\mu) h(\varepsilon)+\lambda_{1} \int_{p_{\min }}^{p} \ell(\varepsilon, x) d x\right\} f(p)$$ where $h$ is the pdf of $ε$, and $\bar{F}(p)=1-F(p)$ is the inverse cdf of $p$.
They then say that this solves as $$\ell(\varepsilon, p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} h(\varepsilon) f(p)$$ where $\kappa_{1}=\lambda_{1} /(\delta+\mu)$, but I can't see how.
Also note that at this time we don't know that if the joint distribution is independent or not because the authors integrate this result to get $\ell(p)=\frac{1+\kappa_{1}}{\left[1+\kappa_{1} \bar{F}(p)\right]^{2}} f(p)$ and thus $\ell(\varepsilon, p)=h(\varepsilon) \ell(p)$ and finally argue that $ε$ and $p$ are uncorrelated.
