Garnier integrable system

In mathematical physics, the Garnier integrable system, also known as the classical Gaudin model is a classical mechanical system discovered by René Garnier in 1917, and solved using abelian integrals on compact Riemann surfaces (algebraic curves) of arbitrarily high genus. It is obtained by taking the 'Painlevé simplification' or 'autonomous limit' of the Schlesinger equations. It may be interpreted as the classical limit of the quantum Gaudin model due to Michel Gaudin. (Similarly, the Schlesinger equations are the classical limit of the Knizhnik–Zamolodchikov equations, expressed in the Heisenberg representation.. )

The Garnier systems were later shown to be of Hamiltonian type , defined on a phase space consisting of the Cartesian product of ${\displaystyle N}$ copies of the dual ${\displaystyle {\mathfrak {gl(r)}}^{*}}$ of the Lie algebra ${\displaystyle {\mathfrak {gl(r)}}^{*}}$, for ${\displaystyle N}$ a positive integer, and completely integrable in the Hamiltonian sense.

They are also a specific case of Hitchin integrable systems, when the algebraic curve that the theory is defined on is the Riemann sphere and the system is tamely ramified.