  The presence of isotropic Lifshitz points for a U(1) symmetric scalar theory
is investigated with the help of the Functional Renormalization Group at the
conjectured lower critical dimension d=4. To this aim, a suitable truncation in
the expansion of the effective action in powers of the field is considered and,
consequently, the Renormalization Group flow is reduced to a set of ordinary
differential equations for the parameters that define the effective action.
Within this approximation, indications of a line of Lifshitz points are found,
that present evident similarities with the properties shown by the line of
fixed points observed in the two dimensional Berezinsky-Kosterlitz-Thouless
phase. In particular, this line of Lifshitz points exhibits the vanishing of
the expectation value of the field, together with a finite stiffness and, for
specific combinations of the parameters that define the effective action, also
the algebraic decay at large distance of the order parameter correlations.
