Galliano VALENT
CNRS Luminy Case 907,
Centre de Physique Th\'eorique,
F-13288 Marseille Cedex 9, FRANCE

Laboratoire de Physique Th\'eorique et des Hautes Energies,
CNRS, Unit\'e associ\'ee URA 280,
2 Place Jussieu, 75251 Paris Cedex 05, FRANCE

Higher conserved quantities for the multi-centre metrics

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. We consider the hamiltonian dynamics generated by the geodesic flow along the U(1) fibers and obtain several  metrics in this class which do exhibit an extra conserved quantity quadratic in the momenta. The corresponding systems are shown to be classically integrable. The relation of these results with some three-dimensional integrable systems with potential is given.