The Nash equilibria of two two-person nonzero-sum differential games with hard constraints on the controls are studied. For both games the open-loop as well as the closed-loop solutions, and their relationships, are discussed. As is well-known for smooth nonzero-sum games, these solutions are generally different. Because of the constraints, the optimal controls are of the bang-bang type, and the solutions of the two problems under consideration are nonsmooth. One deals with non-Lipschitzian differential equations (considering the problem as an optimal control problem for one player while the bang-bang feedback control of the other player is assumed to be fixed), and the corresponding value functions possess singular surfaces. General conditions for the existence and uniqueness of the feedback solutions in this framework are not yet known. It is shown that in the two examples the open-loop and closed-loop solutions differ. As a by-product, the paper aims at a modest exploration of singular surfaces in nonzero-sum games.