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