  In this paper we continue earlier investigations of evolutionary formulations
of the Einstein vacuum constraint equations originally introduced by R\'{a}cz.
Motivated by the strong evidence from these works that the resulting vacuum
initial data sets are generically not asymptotically flat we analyse the
asymptotics of the solutions of a modified formulation by a combination of
analytical and numerical techniques. We conclude that the vacuum initial data
sets generated with this new formulation are generically asymptotically flat.
