  The multi-centre metrics are a family of euclidean solutions of the empty
space Einstein equations with self-dual curvature. For this full class, we
determine which metrics do exhibit an extra conserved quantity quadratic in the
momenta, induced by a Killing-St\" ackel tensor. Our systematic approach brings
to light a subclass of metrics which correspond to new classically integrable
dynamical systems. Within this subclass we analyze on the one hand the
separation of coordinates in the Hamilton-Jacobi equation and on the other hand
the construction of some new Killing-Yano tensors.
