In Part 1 of this paper (Hassan and Amin 1987) a simple algebraic solution for the time-optimal output regulator problem was presented. This solution, which consists of a state feedback control law, has been obtained for all classes of right-invertible decouplable systemsS(A, B, C, E). The results of Part 1 are here extended to all classes of right-invertible systemsS(A, B, C, E). A set of optimal output deadbeat indices (called the ‘optimal set’) is defined and related to the observability indices of the optimal closed-loop system. The time-optimal output regulator problem for a right-invertible non-decouplable systemS(A, B, C, E)is resolved by transformingSinto a decouplable systemSc(A, B, Cc, Ec)having the optimal output deadbeat index σ* ofS. First, an algorithm is presented to construct iteratively, in a well-defined optimal sense, a unimodular left compensatorL(z)and a compensated decouplable systemSc(A, B, Cc, Ec)from the state-space parameters ofS. Then, a family of optimal state feedback matricesFc*, which attains the optimal set ofSc, is given as the optimal solutionF* ofS. For any right-invertible systemS(A, B, C, E), it is shown that the optimal number of control iterations required at most to zero its outputs is equal to the associated uniquely defined μ1index of the right Wiener-Hopf factorization at infinity ofH(z)(the transfer function matrix of the system). A numerical example is worked out to illustrate the design procedures of time-optimal output regulators for non-decouplable systems.