GEODESIC FLOW ON SO(4), KAC-MOODY LIE ALGEBRA AND SINGULARITIES IN THE COMPLEX t-PLANE

The article studies geometrically the Euler-Arnold equations as-sociated to geodesic flow on SO(4) for a left invariant diagonal metric. Such metric were first introduced by Manakov [17] and extensively studied by Mishchenko-Fomenko [18] and Dikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] and Haine [8]. In this problem there are four invariants of the motion defining in C4 = Lie(SO(4) ⊗ C) an affine Abelian surface as complete in-tersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, based on the Kostant-Kirillov coad-joint action. This method allows us to linearizes the problem on a two-dimensional Prym variety Prymσ(C) of a genus 3 Rie-mann surface C. In section 2, the method consists of requiring that the general solutions have the Painleve ́ property, i.e., have no movable singularities other than poles. It was first adopted by Kowalewski [10] and has developed and used more system-atically [3], [4], [8], [13]. From the asymptotic analysis of the differential equations, we show that the linearization of the Euler-Arnold equations occurs on a Prym variety Prymσ(Γ) of an an-other genus 3 Riemann surface Γ. In the last section the Riemann surfaces are compared explicitly. 1. Lie algebra theoretical method Consider the group SO(4) and its Lie algebra so(4) paired with itself, via the customary inner produc