It is well known that finding Nash equilibrium solutions of nonzero-sum differential games is a challenging task. Focusing on a class of linear quadratic differential games, we consider three notions of approximate feedback Nash equilibrium solutions and provide a characterisation of these in terms of matrix inequalities which constitute quadratic feasibility problems. These feasibility problems are then recast first as bilinear feasibility problems and finally as rank constrained optimisation problems, i.e. a class of static problems frequently encountered in control theory.