  Two co-dimensional thick brane-worlds are investigated in quite general terms
for two intersecting scalar fields generating the extra dimension defect. In
general, when one considers two co-dimensional thick brane-worlds, the warp
factor is constructed as a string-like defect. Considering a twofold-warp
factor constructed from two intersecting warp factors, an alternative bulk
configuration is examined. With the brane localization thus driven by two
crossing scalar fields, the possible solvable models obtained from such a two
co-dimensional setup are systematically discussed. The obtained solutions are
classified as five different models organized into two subsets for which some
of their physical properties are evaluated. For models $I$ and $II$, in the
first subset, Einstein equation solutions are rigidly defined, up to some
arbitrary constant. For models $III$, $IV$ and $V$, in the second subset, an
additional degree of freedom not constrained by Einstein equations is admitted.
The solutions are all obtained from a departure statement of assuming a
conformally flat metric for the internal space, which is concomitant to the
proper choice of coordinates. Eventual singularities in the curvature are
identified, however, without affecting the physical appeal of the solutions
described in terms of the stress energy tensor patterns, which are shown to be
free of singularities for model $IV$, besides admitting straightforward
reductions to $(4+1)$-dimensions. In particular, from the framework of models
$III$ and $IV$, one is able to achieve brane-world solutions over two different
geometries of $\mathbb{S}^{2}$ which, as demonstrated, can be reduced to
trivial and non-trivial extensions of the well-known $(4+1)$-dimensional
brane-worlds.
