We identify a class, ℋ, of general Petri net (PN) structures where the existence of a liveness enforcing supervisory policy (LESP) for an instance of this class, initialized at a marking, is sufficient to infer the existence of an LESP when the same instance is initialized at a larger marking. As a consequence, the existence of an LESP for the PN that results when a member of this class is initialized with a marking, isdecidable. Additionally, the maximally permissive LESP, when it exists, can be synthesized using a software tool described in an earlier paper. We also highlight a property that plays a critical role in the speed of convergence of the iterative procedure for the synthesis of the minimally restrictive LESP, when it exits, for any instance of ℋ that uses the aforementioned software package.