Lautenbach's marking condition for liveness (where all minimal siphons are invariant-controlled) has been successfully applied to FMS. However, there are two exceptions with no solutions in literatures. We have uncovered these mysteries and extends the above models to all types of resource-sharing based on the concept of synthesis of minimal siphons. Different structures in the synthesis result in different classes of nets. Each class is maximal in the sense that it covers various classes of nets in the literature. Lautenbach's marking condition for liveness (MCL) is extended to more general cases and we are able to offer a more intuitive insight as to the structural cause for the above exceptions. It also helps to discover new TP-PT generation rules for our proprietary Knitting technique [2-3]. We have studeid each of the above maximal classes and derive its marking condition for liveness. We also propose a procedure to find the S-invariant that controls a minimal siphon.