  In this paper we investigate the parabolic-hyperbolic formulation of the
vacuum constraint equations introduced by R{\'a}cz with a view to constructing
multiple black hole initial data sets without spin. In order to respect the
natural properties of this configuration, we foliate the spatial domain with
2-spheres. It is then a consequence of these equations that they must be solved
as an initial value problem evolving outwards towards spacelike infinity.
Choosing the free data and the "strong field boundary conditions" for these
equations in a way which mimics asymptotically flat and asymptotically
spherical binary black hole initial data sets, our focus in this paper is on
the analysis of the asymptotics of the solutions. In agreement with our earlier
results, our combination of analytical and numerical tools reveals that these
solutions are in general not asymptotically flat, but have a cone geometry
instead. In order to remedy this and approximate asymptotically Euclidean data
sets, we then propose and test an iterative numerical scheme.
