The Last Integrable Case of Kozlov-treshchev Birkhoff Integrable Potentials

The integrability of this system was conjectured in [16] and in the book of V.V. Kozlov [17]. This system appears first in the classification of Birkhoff integrable systems by Kozlov and Treshchev [16]. The classification involves systems with exponential interraction with sufficient number of integrals, polynomial in the momenta. The classification gives necessary conditions for a system with exponential interraction to be Birkhoff integrable. The integrability (or not) of each system in the list was established case by case using various techniques. The only open case which remains is the case of system (1). For this reason, this last case became sort of famous and we refer to it as the last integrable case of Kozlov-Treshchev potentials. We now give a brief historical review of this area, including previous progress in establishing the integrability of Birkhoff integrable systems. We begin with the following general definition which involves systems with exponential interaction: Consider a Hamiltonian of the form