A new canonical form for the minimal realizations of finite dimensional multi-input multi-output linear systems possessing cyclic state spaces has been developed. This minimal realization is called the ‘observable-pair realization’ and may be used as a direct summand of minimal realizations of general linear systems. An algorithm, suitable for computer implementation, is presented for calculating the observable-pair realizations of transfer function matrices of cyclic systems. It is shown that this algorithm is readily used as part of a second algorithm for calculating the minimal realizations of general transfer functions. Finally, it is proved that in a special sense the observable-pair realizations are unique.