  The contribution of this paper is twofold. First, we generalize the
definition of discrete isothermic surfaces. Compared with the previous ones, it
covers more discrete surfaces, e.g., the associated families of discrete
isothermic minimal and non-zero constant mean curvature (CMC in short)
surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we
show that the discrete isothermic CMC surfaces can be obtained by the discrete
holomorphic data (a solution of the additive rational Toda system) via the
discrete generalized Weierstrass type representation.
