  The present paper deals with the nonhomogeneous incompressible asymmetric
fluids equations in dimension $d= 2,3$. The aim is to prove the unique global
solvability of the system with only bounded nonnegative initial density and
$H^{1}$ initial velocities. We first construct the global existence of the
solution with large data in 2-D. Next, we establish the existence of local in
time solution for arbitrary large data and global in time for some smallness
conditions in 3-D. Finally, the uniqueness of the solution is proved under
quite soft assumptions about its regularity through a Lagrangian approach. In
particular, the initial vacuum is allowed.
