  We investigate the existence and uniqueness issues of the 3D incompressible
Hall-magnetohydrodynamic system supplemented with initial velocity $u_0$ and
magnetic field $B_0$ in critical regularity spaces.In the case where $u_0,$
$B_0$ and the current $J_0:=\nabla\times B_0$ belong to the homogeneous Besov
space $\dot B^{\frac 3p-1}_{p,1},$ $\:1\leq p<\infty,$ and are small enough, we
establish a global result and the conservation of higher regularity.If the
viscosity is equal to the magnetic resistivity, then we obtain the global
well-posedness provided $u_0,$ $B_0$ and $J_0$ are small enough in the
\emph{larger} Besov space $\dot B^{\frac12}_{2,r},$ $r\geq1.$If $r=1,$ then we
also establish the local existence for large data, and exhibit continuation
criteria for solutions with critical regularity. Our results rely on an
extended formulation of the Hall-MHD system, that has some similarities with
the incompressibleNavier-Stokes equations.
