For a given system (A;B) and a subspace S, the Cover Problem consits of ¯nding all (A;B)-invariant subspaces
containing S. For controllable systems, the set of these subspaces can be suitably strati¯ed. In this paper, necessary and
su±cient conditions are given for the cover problem to have a solution on a given strata. Then the geometry of these solutions
is studied. In particular, the set of the solutions is provided with a di®erentiable structure and a parametrization of all solutions
is obtained through a coordinate atlas of the corresponding smooth manifold.