  U(1) symmetries play a central role in constructing phenomenologically viable
F-theory compactifications that realize Grand Unified Theories (GUTs). In
F-theory, gauge symmetries with abelian gauge factors are modeled by singular
elliptic fibrations with additional rational sections, i.e. a non-trivial
Mordell-Weil rank. To determine the full scope of possible low energy theories
with abelian gauge factors, which allow for an F-theory realization, it is
central to obtain a comprehensive list of all singular elliptic fibrations with
extra sections. We answer this question for the case of one abelian factor by
applying Tate's algorithm to the elliptic fiber realized as a quartic in the
weighted projective space P^{(1,1,2)}, which guarantees, in addition to the
zero section, the existence of an additional rational section. The algorithm
gives rise to a tree-like enhancement structure, where each fiber is
characterized by a Kodaira fiber type, that governs the non-abelian gauge
factor, and the separation of the two sections. We determine Tate-like forms
for elliptic fibrations with one extra section for all Kodaira fiber types. In
addition to standard Tate forms that are determined by the vanishing order of
the coefficient sections in the quartic (so-called canonical models),the
algorithm also gives rise to fibrations that require non-trivial relations
among the coefficient sections. Such non-canonical models have
phenomenologically interesting properties, as they allow for a richer charged
matter content, and thus codimension two fiber structure, than the canonical
models that have been considered thus far in the literature. As an application
we determine the complete set of codimension one fibers types, matter spectra,
both canonical and non-canonical, for SU(5) x U(1) models.
