ar X iv : s ol v - in t / 9 60 30 04 v 1 1 2 M ar 1 99 6 Lax representation for two – particle dynamics splitting on two tori

Lax representation in terms of 2 × 2 matrices is constructed for separable multiply–periodic systems splitting on two tori. Hyperelliptic Kleinian functions and their reduction to elliptic functions are used. Introduction. Completely integrable systems with two degree of freedoms and with dynamics splitting on two tori have been largely investigated during the past years as examples of separable multiply-periodic systems. The list of such systems includes the well known in-tegrable cases of the Hénon-Heiles system [1], several integrable cases of quartic potentials [2], the motion of a particle in the Coulomb potential and in external uniform field, the Chaplygin top [3], etc. A Lax representation for these systems can be readily constructed in terms of a direct product of Lax operators [1], one for each splitting tori, as first proposed in ref.[4]. This approach leads for a system of two