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Using the theory of $1+1$ hyperbolic systems we put in perspective themathematical and geometrical structure of the celebrated circularly polarizedwaves solutions for isotropic hyperelastic materials determined by Carroll inActa Mechanica 3 (1967) 167--181. We show that a natural generalization of thisclass of solutions yields an infinite family of \emph{linear} solutions for theequations of isotropic elastodynamics. Moreover, we determine a huge class ofhyperbolic partial differential equations having the same property of the shearwave system. Restricting the attention to the usual first order asymptoticapproximation of the equations determining transverse waves we provide thecomplete integration of this system using generalized symmetries.

On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics