Galliano Valent
2 Place Jussieu, 75251 Paris Cedex 05, France
Laboratoire de Physique Mathématique de Provence
Publications:
Valent G.
Superintegrable Models on Riemannian Surfaces of Revolution with Integrals of any Integer Degree (I)
2017, vol. 22, no. 4, pp. 319352
Abstract
We present a family of superintegrable (SI) systems which live on a Riemannian
surface of revolution and which exhibit one linear integral and two integrals of any integer
degree larger or equal to 2 in the momenta. When this degree is 2, one recovers a metric due
to Koenigs.
The local structure of these systems is under control of a linear ordinary differential equation of
order $n$ which is homogeneous for even integrals and weakly inhomogeneous for odd integrals.
The form of the integrals is explicitly given in the socalled “simple” case (see Definition 2).
Some globally defined examples are worked out which live either in $\mathbb{H}^2$ or in $\mathbb{R}^2$.

Valent G.
Global Structure and Geodesics for Koenigs Superintegrable Systems
2016, vol. 21, no. 5, pp. 477509
Abstract
We present a new derivation of the local structure of Koenigs metrics using a framework laid down by Matveev and Shevchishin. All of these dynamical systems allow for a potential preserving their superintegrability (SI) and most of them are shown to be globally defined on either ${\mathbb R}^2$ or ${\mathbb H}^2$. Their geodesic flows are easily determined thanks to their quadratic integrals. Using Carter (or minimal) quantization, we show that the formal SI is preserved at the quantum level and for two metrics, for which all of the geodesics are closed, it is even possible to compute the classical action variables and the point spectrum of the quantum Hamiltonian.

Valent G.
On a Class of Integrable Systems with a Quartic First Integral
2013, vol. 18, no. 4, pp. 394424
Abstract
We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some detail and leads to a class of models on the manifolds $\mathbb{S}^2$, $\mathbb{H}^2$ or $\mathbb{R}^2$. As special cases we recover Kovalevskaya’s integrable system and a generalization of it due to Goryachev.
