  We explore solutions of six dimensional gravity coupled to a non-linear sigma
model, in the presence of co-dimension two branes. We investigate the
compactifications induced by a spherical scalar manifold and analyze the
conditions under which they are of finite volume and singularity free. We
discuss the issue of single-valuedness of the scalar fields and provide some
special embedding of the scalar manifold to the internal space which solves
this problem. These brane solutions furnish some self-tuning features, however
they do not provide a satisfactory explanation of the vanishing of the
effective four dimensional cosmological constant. We discuss the properties of
this model in relation with the self-tuning example based on a hyperbolic sigma
model.
