For a given system S(A, B) and a subspace S, the cover problem consists of finding all (A, B)-invariant subspaces containing S. For controllable systems, the set of these subspaces can be suitably stratified. In this paper, necessary and sufficient 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 differentiable structure and a parameterization of all solutions is obtained through a coordinate atlas of the corresponding smooth manifold.