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This paper is on several basic properties of term rewrite systems: reachability, joinability, uniqueness of normal forms, unique normalization, confluence, and existence of normal forms, for subclasses of rewrite systems defined by syntactic restrictions on variables. All these properties are known to be undecidable for the general class and decidable for ground (variable-free) systems. Recently, there has been impressive progress on efficient algorithms or decidability results for many of these properties. The aim of this paper is to present new results and organize existing ones to clarify further the boundary between decidability and undecidability for these properties. Another goal is to spur research towards a complete classification of these properties for subclasses defined by syntactic restrictions on variables. The proofs of the presented results may be intrinsically interesting as well due to their economy, which is partly based on improved reductions between some of the properties.