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We construct the generalised Eigenfunctions of the entries of the monodromymatrix of the $N$-site modular XXZ magnet and show, in each case, that theseform a complete orthogonal system in $L^2(\mathbb{R}^N)$. In particular, wedevelop a new and simple technique, allowing one to prove the completeness ofsuch systems. As a corollary of out analysis, we prove the Bystko-Teschnerconjecture relative to the structure of the spectrum of the $\boldsymbol{\texttt{B} }(\la)$-operator for the odd length lattice Sinh-Gordon model.

On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models