Unlike traditional classification by output conditions of places, synchronized choice nets (SNCs) were defined as a new class of nets where all first-order structures (FOSs) are symmetric. SNCs are extended to deal with nets with asymmetric first-order structures (AFOSs). A subclass called extended SNCs (ESNCs) can be converted to a general Petri net, called a weighted SNC (WSNC). A WSNC is an SNC if all its arc weights are one. We apply existing theory to check if a WSNC is structural live and bounded. If it is, we derive the sufficient and necessary marking condition (the weighted marking $W(M_0)>0$) for liveness. Thus, it is live when invariant-controlled. We then enhance it in three aspects and apply it to flexible manufacturing systems with resource sharing. Extensions to ESNCs are also discussed.