From the point of view of the theory of the partial differential equations, the paper is concerned with the Helmholtz version of the Grushin equation \[ (\partial^2_x+ x^{2r}\partial^2_y) u+ \rho^2u= 0,\quad r=1,2,\dots, \] which, by separation of variables, reduces to the analysis of eigenfunctions of anisotropic harmonic oscillators. Classical results in this connection are revisited and improved, with attention to the geometry of the related geodesic curves, in the frame of the sub-Riemannian geometry, see for example [\textit{R. Montogomery} [A tour of sub-Riemannian geometries, their geodesics and applications. Providence, RI: AMS (2002; Zbl 1044.53022)]. The main issue of the paper is represented by the applications to super-dimensional metamaterials. An interesting implementation is given in the last sections.