view. His strategy is based on a perturbation expansion in the parameter $p=\exp(-\alpha r)$. (In this connection it should be pointed out that $H(g;x)$ with $g_2=g_3=0$ reduces to $H_t(g_0,g_1;x)$ for $\alpha\to\infty$.)

The main result of Theorem \[theorem2.6\] is that for $g\in\Pi_r$ the spectra of $H(g)$ and $H(g')$ coincide. Unfortunately, we have not been able to show that the singular values in (\[sval\]) are distinct for all $g\in\Pi_r$, a feature that seems plausible to us. If this is indeed the case, then it would follow that $e_n(g)$ is the $H(g)$-eigenvector with eigenvalue $E_n(g)$ in the obvious eigenvalue list $$\begin{gathered}
E_0(g)<E_1(g)<E_2(g)<\cdots,\qquad g\in\Pi_r.\end{gathered}$$

In Section \[section3\] we fix $g\in\Pi_r$ and address the question for which $\hat{g}\in\tilde{\Pi}$ the spectrum of $H(\hat{g})$ coincides with that of $H(g)$. From Theorem \[theorem2.6\] we know this is true for $\hat{g}=g'$, but there are a few more involutory $g$-transformations for which this is valid. We determine the group $G$ generated by these involutions in Theorem \[theorem3.1\]. It leaves the parameter set $$\begin{gathered}
\label{PiG}
\Pi_G\equiv \{ g\in\Pi_r\mid g_0+g_1+2>g_2+g_3\}\end{gathered}$$ invariant, and is isomorphic to the permutation group $S_4$. (For ease of reference, we collect the four parameter sets (\[PiG\]), (\[Pir\]), (\[Pi\]) and (\[tPi\]) in a chain $$\begin{gathered}
\Pi_G\subset \Pi_r\subset \Pi\subset\tilde{\Pi},\end{gathered}$$ and 