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Petri nets proved useful to describe various real-world systems, but many of their properties are very hard to check. To alleviate this difficulty, subclasses are often considered. The class of weighted marked graphs with relaxed place constraint (WMG=< for short), in which each place has at most one input and one output, and the larger class of choice-free (CF) nets, in which each place has at most one output, have been extensively studied to this end, with various applications. In this work, we develop new properties related to the fundamental and intractable problems of reachability, liveness and reversibility in weighted Petri nets. We focus mainly on the homogeneous Petri nets with a single shared place (H1S nets for short), which extend the expressiveness of CF nets by allowing one shared place (i.e. a place with at least two outputs and possibly several inputs) under the homogeneity constraint (i.e. all the output weights of the shared place are equal). Indeed, this simple generalization already yields new challenging problems and is expressive enough for modeling existing use-cases, justifying a dedicated study. One of our central results is the first characterization of liveness in a subclass of H1S nets more expressive than WMG=< that is expressed by the infeasibility of an integer linear program (ILP) of polynomial size. This trims down the complexity to co-NP, contrasting with the known EXPSPACE-hardness of liveness in the more general case of weighted Petri nets. In the same subclass, we obtain a new reachability property related to the live markings, which is a variant of the well-known Keller's theorem. Another central result is a new reversibility characterization for the live H1S class, simplifying its checking. Finally, we apply our results to use-cases, highlight their scalability and discuss their extensibility to more expressive classes.