On the Calogero-Moser solution by root-type Lax pair (Q2862903)

The rational Calogero-Moser system is a system of \(n\) particles on the line with governing Hamiltonian \(H = \frac{1}{2} \sum_{s=1}^n p_s^2 + \sum_{1\leq i < j \leq n} (q_i-q_j)^{-2}\), where \(q_i\) is the position of the \(i^{\mathrm{th}}\) particle and \(p_i\) its momenta. There exists a Lax pair for the corresponding Hamiltonian system and the positions \(q_i=q_i(t)\) at time \(t\) are given by the eigenvalues of the \((n \times n)\)-matrix NEWLINE\[NEWLINE \left( \begin{matrix} q_1(0)+tp_1(0) & \dots & \frac{t}{|q_1(0)-q_n(0)|} \\ \dots & \dots & \dots \\ \frac{t}{|q_n(0)-q_1(0)|} & \dots & q_n(0)+tp_n(0) \end{matrix} \right). NEWLINE\]NEWLINE For \(t=0\), let the coordinates satisfy \(\sum q_i = \sum p_i =0\) and \((*)\;q_1<q_2< \cdots < q_n\). Remark that \(H\) is a positive constant and so \(|q_i(t)-q_j(t)|>\frac{1}{\sqrt{H}}\), which implies that the order (*) is preserved for all \(t \in \mathbb{R}\). There exists a well-known connection between the vector \(q=(q_1,\dots,q_n)\) and the root system \(\Phi(A_{n-1})\) of the Lie algebra \(A_{n-1}\); we know the Euclidean scalar products \((\alpha,q)\) for each root \(\alpha \in \Phi(A_{n-1})\). Could one restore \(q\) from a fixed set \(\{ (\alpha,q),\, \alpha \in \Phi(A_{n-1}) \}\)? The author gives a negative answer: both \(q =\{-28,-22,-16,8,20,38\}\) and \(q'=\{-34,-28, 2, 8,20,32\}\) have the same \(\{ (\alpha,q),\, \alpha \in \Phi(A_5) \} = \pm \{ 6,6,12,12,18,24,30,30,36,36,42,48,54,60,66\}\). This counterexample was produced by computer search.