A zero-sum differential game of infinite horizon is considered. Positive switching costs are associated with each player. We prove that under a condition, which is different from the Isaacs' condition, the Elliot-Kalton value of the game always exists. The value is the unique viscosity solution of corresponding Isaacs' type system of equations, which appears to be a system of quasi-variational inequalities with bilateral obstacles. The discussion of the case that the switching costs approach to zero shows that the Isaacs' condition still has to be assumed to guarantee that the limit of the values corresponding to positive switching costs converge to the value corresponding to zero switching costs—the classical Elliot-Kalton value of the game.