  We consider (d_0+2)-dimensional configurations with global strings in the two
extra dimensions and a flat metric in d_0 dimensions, endowed with a warp
factor e^{2\gamma} depending on the distance l from the string center. All
possible regular solutions to the field equations are classified by the
behavior of the warp factor and the extra-dimensional circular radius r(l).
Solutions with r\to \infty and r\to \const >0 as l\to \infty are interpreted in
terms of thick brane world models. Solutions with r \to 0 as l \to l_c > 0,
i.e., those with a second center, are interpreted as either multi-brane systems
(which is appropriate for large enough distances l_c between the centers) or as
Kaluza-Klein type configurations with extra dimensions invisible due to their
smallness. For the case of the Mexican-hat symmetry breaking potential, we
build a full map of regular solutions on the (\epsilon,\Gamma) parameter plane,
where \epsilon acts as an effective cosmological constant while \Gamma
characterizes the strength of gravity. The trapping properties of candidate
brane worlds for test scalar fields are discussed. Good trapping properties for
massive fields are found for models with growing warp factors. Kaluza-Klein
type models are shown to possess nontrivial warp factor behaviors, leading to
matter particle mass spectra which seem promising from the standpoint of
hierarchy problems.
