Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of ( ) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.