For a bounded domain $\Omega \subset {ℝ}^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u+u·\nabla u+\nabla p=f$, $divu=k$, $u_{{|}_{\partial \Omega }}=g$ with $u\in L^q$, $q\ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of...