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Existence of acoustic waves guided by the tip of anisotropic wedges is studied.Criteria for their existence in certain rectangular wedges are derived.These criteria are based on surface acoustic wave properties.The criteria have been verified by numerical calculations.The existence of acoustic waves with displacements localized at the tip of an isotropic elastic wedge was rigorously proven by Kamotskii, Zavorokhin and Nazarov. This proof, which is based on a variational approach, is extended to rectangular anisotropic wedges. For two high-symmetry configurations of rectangular edges in elastic media with tetragonal symmetry, a criterion is derived that allows identifying the boundary between the regions of existence for wedge modes of even and odd symmetry in regions of parameter space, where even- and odd-symmetry modes do not exist simultaneously. Furthermore, rectangular edges with non-equivalent surfaces are analyzed, and it is shown that at rectangular edges of cubic elastic media with one (1 1 0) surface and one (0 0 1) surface, a tip-localized guided wave always exists, apart from special cases that are characterized.