Petri nets are useful in modeling and analyzing various types of discrete-event systems such as parallel processing systems, distributed systems, and sequential control systems, because Petri nets can easily be used to represent such properties of these systems as concurrency, nondecidability, and causality. Various behavioral analytic problems on Petri nets are reduced to reachability and liveness on them. It is also known that the decidability of liveness is equivalent to that of reachability which is solvable. However, useful necessary and sufficient structural liveness conditions have been given only for extended free-choice (EFC) nets and their subclasses. Moreover recently, a necessary and sufficient structural liveness condition for a useful subclass NKT=(SKT, TKT, FKT, MoKT) (i.e., a Petri net in which each minimal structural deadlock (MSDL) contains at least one real or virtual kindling trap, each locally structural-live MSDL ND=(SD, TD, FD, MoD) is never globally dead even if all key transitions for local liveness of each MSDL are controlled by the net of SKTSD s.t. SKTSD, and there exists no singular MSDL of type (α)) has also been given. In this paper, in order to give one of the bases for a necessary and sufficient "structural" or "initial-marking-based" liveness condition for a general Petri net N, we will, first, directly present a necessary and sufficient local liveness condition for each MSDL with a real deadlock-trap structure in a subclass Ñ (N) using the net structure and initial token distribution and extending basic concepts used in NKT, where Ñ is a general Petri net without live behavioral traps, local liveness means a useful necessary condition for the above final goal, and real deadlock-trap structure means that each MSDL in Ñ contains at least one minimal structural trap. Secondly, a new subclass is shown in which, if the above locally structural liveness condition for each MSDL holds, then the whole-net liveness is also guaranteed. It is also argued that the obtained results are applicable to describing new live behavioral traps and deriving a necessary and sufficient structural liveness condition, which is the final goal in this work, for a general Petri net N.