  We present solutions in six-dimensional gravity coupled to a sigma model, in
the presence of three-brane sources. The space transverse to the branes is a
compact non-singular manifold. The example of O(3) sigma model in the presence
of two three-branes is worked out in detail. We show that the four-dimensional
flatness is obtained with a single condition involving the brane tensions,
which are in general different and may be both positive, and another
characteristic of the branes, vorticity. We speculate that the adjustment of
the effective four-dimensional cosmological constant may occur through the
exchange of vorticity between the branes. We then give exact instanton type
solutions for sigma models targeted on a general K\"ahler manifold, and
elaborate in this framework on multi-instantons of the O(3) sigma model. The
latter have branes, possibly with vorticities, at the instanton positions, thus
generalizing our two-brane solution.
