||The article studies geometrically the Euler-Arnold equations associatedto geodesic flow on SO(4) for a left invariant diagonal metric. Such metric were first introduced by Manakov  and extensively studied by Mishchenko-Fomenko  andDikii . An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke  andHaine . In this problem there are four invariants of the motion defining in C4 = Lie(SO(4) ⊗ C) an affine Abelian surface as complete intersection of four quadrics. The first section is devoted to a Lie algebra theoretical approach, basedon the Kostant-Kirillov coadjoint action. This methodallo ws us to linearizes the problem on a two-dimensional Prym variety Prymσ(C) of a genus 3 Riemann surface C. In section 2, the methodconsists of requiring that the general solutions have the Painlev´e property, i. e. , have no movable singularities other than poles. It was first adopted by Kowalewski  andhas developedandusedmore systematically , , , . From the asymptotic analysis of the differential equations, we show that the linearization of the Euler- Arnoldequations occurs on a Prym variety Prymσ(Γ) of an another genus 3 Riemann surface Γ. In the last section the Riemann surfaces are comparedexplicitly .