The Henon-Heiles differential system and Prym varieties (Q1880100)

The author considers the following Hénon-Heiles system \(\dot{q}_1=p_1\), \(\dot{q}_2=p_2\), \(\dot{p}_1=-Aq_1-2q_1q_2\), \(\dot{p}_2=-Bq_2-q_1^2-6q_2^2\), where \(A\) and \(B\) are constants. He shows (Theorem 2) that the generic fiber \(F\) over invariant varieties is an affine part of an abelian surface \(\widetilde{F}\). Thus, this Hénon-Heiles system is algebraically completely integrable. It is also proved (Theorem 3) that \(\widetilde{F}\) can be identified with the dual of the Prym variety of a certain double covering \(\Gamma/\Gamma_0\). The author describes explicitely the hyperelliptic curve \(\Gamma\) whose involution generates this covering over an elliptic curve \(\Gamma_0\).