  We show that six-dimensional supergravity models admit nonsingular solutions
in the presence of flat three-brane sources with positive tensions. The models
studied in this paper are nonlinear sigma models with the target spaces of the
scalar fields being noncompact manifolds. For the particular solutions of the
scalar field equations which we consider, only two brane sources are possible
which are positioned at those points where the scalar field densities diverge,
without creating a divergence in the Ricci scalar or the total energy. These
solutions are locally invariant under 1/2 of D=6 supersymmetries, which,
however, do not integrate to global Killing spinors. Other branes can be
introduced by hand by allowing for local deficit angles in the transverse space
without generating any kind of curvature singularities.
