This paper gives a complete characterization of all optimal linear quadratic static state feedback controllers if the system is described by an index one descriptor system Ex(t) = Ax(t) + Bu(t). No definiteness restrictions are made with respect to the quadratic performance criterion. The explicit characterization of all solutions is achieved by transforming to a coordinate system, which can be computed by performing a singular value decomposition of E. This enhances the formulation of a numerical algorithm to calculate the solutions. Based on this explicit characterization of the solution set, we can characterize the set of all linear feedback Nash equilibria, in the context of non-cooperative linear quadratic differential games. The results are illustrated in an example.