Abstract <p>The existence of surface acoustic waves (SAWs) in a semi-infinite substrate with a deposited solid layer is theoretically investigated. The substrate and the layer are not piezoelectrics but may belong to any class of crystallographic symmetry. By writing down the dispersion equation in the form of a condition on the substrate and layer impedance matrices, it is possible to determine, using the properties of impedances, the maximum allowable number of surface waves, depending on the contact type and the relation between the velocities of bulk waves in the substrate and layer materials. A dispersion equation is derived for the symmetric orientation of an orthorhombic substrate with a deposited monoatomic layer, and the possibility of existence of a purely flexure surface acoustic wave in the case of a very hard surface layer, for example, a graphene monolayer on a soft polymer substrate, is shown.</p>