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In our previous publications (IJTAF 2019, Math. Finance 2020), we introduced a general class of SINH-regular processes and demonstrated that efficient numerical methods for the evaluation of the Wiener-Hopf factors and various probability distributions (prices of options of several types) in Lévy models can be developed using only a few general properties of the characteristic exponent $ψ$. Essentially all popular Lévy processes enjoy these properties. In the present paper, we define classes of Stieltjes-Lévy processes (SL-processes) as processes with completely monotone Lévy densities of positive and negative jumps, and signed Stieltjes-Lévy processes (sSL-processes) as processes with densities representable as differences of completely monotone densities. We demonstrate that 1) all crucial properties of $ψ$ are consequences of the representation $ψ(ξ)=(a^+_2ξ^2-ia^+_1ξ)ST(\cG_+)(-iξ)+(a^-_2ξ^2+ia^-_1ξ)ST(\cG_-)(iξ)+(\sg^2/2)ξ^2-iμξ$, where $ST(\cG)$ is the Stieltjes transform of the (signed) Stieltjes measure $\cG$ and $a^\pm_j\ge 0$; 2) essentially all popular processes other than Merton's model and Meixner processes areSL-processes; 3) Meixner processes are sSL-processes; 4) under a natural symmetry condition, essentially all popular classes of Lévy processes are SL- or sSL-subordinated Brownian motion.