Duality in integrable systems and generating functions for new hamiltonians

Duality in the integrable systems arising in the context of Seiberg–Witten theory shows that their tau-functions indeed can be seen as generating functions for the mutually Poisson-commuting hamiltonians of the dual systems. We demonstrate that the Θ-function coefficients of their expansion can be expressed entirely in terms of the co-ordinates of the Seiberg–Witten integrable system, being, thus, some set of hamiltonians for a dual system.