The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters

It is shown that the KNS-spectral measure of the typical Schreier graph of the action of $3$-generated $2$-group of intermediate growth constructed by the first author in 1980 on the boundary of binary rooted tree coincides with the Kesten’s spectral measure, and coincides (up to affine transformation of $\mathbb R$) with the density of states of the corresponding diatomic linear chain.
Jacoby matrix associated with Markov operator of simple random walk on these graphs is computed. It shown shown that KNS and Kesten's spectral measures of the Schreier graph based on the orbit of the point $1^{\infty}$ are different but have the same support and are absolutely continuous with respect to the Lebesgue measure.