On Rosenhain–Göpel Configurations and Integrable Systems

We give a birational morphism between two types of genus 2 Jacobians in $\mathbb{P}^{15}$. One of them is related to an Algebraic Completely Integrable System: the Geodesic Flow on $SO(4)$, metric II (so termed after Adler and van Moerbeke). The other Jacobian is related to a linear system in $|4\Theta|$ with 12 base points coming from a Göpel tetrad of 4 translates of the $\Theta$ divisor. A correspondence is given on the base spaces so that the Poisson structure of the $SO(4)$ system can be pulled back to the family of Göpel Jacobians.