The Garnier and Hénon-Heiles potentials and the Toda lattice

Abstract

The last chapter is devoted to three other integrable Hamiltonian system which are of interest, especially from the point of view of algebraic geometry. We study in Section 2 an integrable quartic potential on the plane, which was discovered by Gamier as a very special case of a large family of integrable systems which he derived from the Schlesinger equations (see [Gar]); our results were first published in [Van3]. In Paragraph 2.1 we will show in two different ways that the general fiber of its moment map is an affine part of an Abelian surface of type (1,4). One method uses some of their specific geometry, as given in the beautiful paper [BLvS], the other one is based on a morphism to the odd master system. It shows that the Garnier potential is an a.c.i. system of type (1,4) (an a.c.i. system of this type was not known before). Moreover the morphism to the odd master system leads to a Lax representation of the quartic potential. In Paragraph 2.5 we consider the limiting case in which the potential is a central potential. Then the general fiber of the moment map is not an affine part of an Abelian variety but is a C* bundle over an elliptic curve.