Symmetries of the Garnier system and of the associated polynomial Hamiltonian system
For a completely integrable Pfaffian system \[ E(\theta):\;dx_ i=\sum^{n}_{j=1}G_{ij}(x,t,\theta)dt_ j,\quad G_{ij}\in {\mathbb{C}}(x,t),\quad i=1,...,m \] depending on parameters \(\theta \in {\mathbb{C}}^ N\), a symmetry for E(\(\theta\)) is a pair (S,l) of a birational transformation S: (x,t)\(\to (x',t')\) and an affine transformation l: \({\mathbb{C}}^ N\to {\mathbb{C}}^ N\) such that \((S^{- 1})^*E(\theta)=E(l(\theta))\). In this paper, a completely integrable Hamiltonian system \({\mathcal G}_ n\), with several time variables, and its associated Hamiltonian system \({\mathcal H}_ n\) are considered. The system \({\mathcal G}_ n\) is obtained by the monodromy preserving deformation of the second order Fuchsian equation on \({\mathbb{P}}^ 1\) and \({\mathcal H}_ n\) is derived from \({\mathcal G}_ n\) by a rational symplectic transformation. For the systems \({\mathcal G}_ n\) and \({\mathcal H}_ n\), a group of symmetries isomorphic to the Weyl group of type \(B_{n+3}\) is constructed.
completely integrable Hamiltonian system
scientific article; zbMATH DE number 4199084
scientific article