Fibration of the phase space of the periodic Toda lattice (Q1900607)

The Toda lattice equation of the view \[ \dot Q_n = P_n,\;\dot P_n = e^{Q_{n-1} - Q_n} - e^{Q_n - Q_{n+1}} \] is studied. Introducing the Flaschka coordinates \(b_n= -P_n\), \(a_n = e^{(Q_n - Q_{n+1})/2}\) there is considered the following set \(M\) as phase space for such equation: \[ M= \left\{(b,a) \in\mathbb{R}^{2N} \left|\sum^N_{j=1} b_j= 0,\;a_j > 0,\;\prod^N_{j=1} a_j = 1 \right.\right\}. \] \(M\) is a symplectic manifold when endowed with the symplectic structure provided by some symplectic matrix \(J\). To describe the fibration of \(M\) by isospectral sets the authors introduce a model space \({\mathcal M}\) by \[ {\mathcal M}: =\bigl\{R = (R_k)_{1\leq k \leq N-1} \mid R_k \text{ is a symmetric, real, trace -- free matrix} \bigr\} \] and they set \(\text{Iso} (R) = \{S\in {\mathcal M} \mid \text{Spec} S_k = \text{Spec} R_k\), \(1\leq k\leq N-1\}\). The main result of the paper is the following theorem. Theorem. There exists a map \(\Phi: M\to {\mathcal M}\) such that (i) \(\Phi\) is \(1:1\) and onto, (ii) \(\Phi\) and its inverse are real analytic, (iii) \(\Phi (\text{Iso} (b,a)) = \text{Iso} (\Phi (b,a))\).