Spaces of initial conditions of Garnier system and its degenerate systems in two variables
Here, the spaces of initial conditions of Garnier system \(\mathcal H_J\), \(J=11111\) and its degenerate systems \(\mathcal H_J\), \(J=1112\), 113, 122, 14, 23, 5, in two variables [see \textit{H. Kimura}, Ann. Mat. Pura Appl. IV. Ser. 155, 25--74 (1989; Zbl 0693.34043)] are constructed. These systems are completely integrable Hamiltonian systems of degree 2 of the form \(dq_k=\sum_{i=1,2}\frac{\partial H_{Ji}}{\partial p_k}ds_i\), \(dp_k=-\sum_{i=1,2}\frac{\partial H_{Ji}}{\partial q_k}ds_i\), \(k=1,2\). The Hamiltonians for all \(J\) are certain polynomials of \(q_1\), \(q_2\), \(p_1\), \(p_2\) whose coefficients are rational functions of \(s=(s_1,s_2)\) holomorphic in a domain \(B_{J}\subset \mathbb C^2\). These systems are obtained as monodromy preserving deformations with regular or irregular singular points and apparent singular points. The author describes these systems as symplectic manifolds and expresses them as polynomial Hamiltonian systems on all affine charts.
scientific article; zbMATH DE number 5119782
scientific article