  A new approach to building models of generalized parton distributions (GPDs)
is discussed that is based on the factorized DD (double distribution) Ansatz
within the single-DD formalism. The latter was not used before, because
reconstructing GPDs from the forward limit one should start in this case with a
very singular function f({\beta})/{\beta} rather than with the usual parton
density f({\beta}). This results in a non-integrable singularity at {\beta}=0
exaggerated by the fact that f({\beta})'s, on their own, have a singular
{\beta}^{-a} Regge behavior for small {\beta}. It is shown that the singularity
is regulated within the GPD model of Szczepaniak et al., in which the Regge
behavior is implanted through a subtracted dispersion relation for the
hadron-parton scattering amplitude. It is demonstrated that using proper
softening of the quark-hadron vertices in the regions of large parton
virtualities results in model GPDs H(x,{\xi}) that are finite and continuous at
the "border point" x={\xi}. Using a simple input forward distribution, we
illustrate implementation of the new approach for explicit construction of
model GPDs. As a further development, a more general method of regulating the
{\beta}=0 singularities is proposed that is based on the separation of the
initial single DD f({\beta}, {\alpha}) into the "plus" part
[f({\beta},{\alpha})]_{+} and the D-term. It is demonstrated that the "DD+D"
separation method allows to (re)derive GPD sum rules that relate the difference
between the forward distribution f(x)=H(x,0) and the border function H(x,x)
with the D-term function D({\alpha}).
