We introduce a noncanonical ("new-time") transformation which exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability. In this way we can construct new integrable systems and, for example, explain the observed duality between the Henon-Heiles and Holt models. It is shown that the transformation can sometimes connect weak-and full-Painleve Hamiltonians. We also discuss quantum integrability and find the origin of the deformation -^-# 2 x~2.PACS numbers: 03.20. + i, 02.30. +gThe search for (and the discovery of) integrable dynamical systems is a most fascinating branch of nonlinear physics, one which has been the center of intensive activity in the past decade. 1 Integrable systems are quite rare and still only a few examples are known. In this paper we will present a novel transformation that relates integrable Hamiltonian systems.Several methods have been devised for the investigation of integrability. One method that has met with particular success in the last few years is singularity analysis, which associates integrability with the Painleve property, i.e., a movable polelike singularity (t-t 0 )~" in the solution of the equations of motion. It was used a century ago by Kowalevskaya, 2 who identified with it the last integrable configuration of the heavy top. The method was resurrected by Ablowitz, Ramani, and Segur 3 and by now several works which have combined Painleve analysis with explicit construction of constants of motion have yielded new integrable systems (see, e.

No abstract

The discovery of new integrable two-dimensional Hamiltonian systems is reported. The analytic structure of the solutions makes necessary the generalization of the Painleve conjecture, a widely used integrability criterion. Such a generalization is presented, which the authors believe should replace the usual conjecture for two-dimensional Hamiltonian systems. It is indeed compatible with all the systems already found and, in addition, leads to still new integrable cases.

No abstract