Tri-Hamiltonian extensions of separable systems in the plane

A method to construct tri-Hamiltonian extensions of a separable system is presented. The procedure is tested for systems, with a Hamiltonian quadratic in the momenta, separable in classical sense in any of the four sets of orthogonal separable coordinates on the Euclidean plane. Some explicit examples are constructed. Finally a conjecture on possible generalizations to other classes of systems is discussed: in particular, the method can be adapted to the 11 orthogonal separable coordinate sets of the Euclidean three-space. (c) 2006 American Institute of Physics.