Summary: We address the problem of scheduling a set of generic tasks to be performed infinitely often without preemption on finitely many parallel processors. These tasks are subject to a set of uniform constraints, modeled by a uniform graph \(G\). The problem is to find a periodic schedule that minimizes the cycle time. Although the problem is NP-hard, we show that the special case where \(G\) is a circuit can be solved in polynomial time. We then study the structure of the set of solutions in the general case and determine several dominant subsets of schedules by analyzing the periodic structure of the allocation function and the induced uniform constraints on the schedule. In particular, we prove the dominance of circular schedules, in which the occurrences of any task are successively performed by all the processors. Finally, we propose a greedy heuristic that provides a good circular schedule.