The algebraic complete integrability of geodesic flow onSO(N)

We study for which left invariant diagonal metrics λ onSO(N), the Euler-Arnold equations $$dot X = [x,lambda (X)], X = (x_{ij} ) in so(N), lambda _{ij} x_{ij} , lambda _{ij} = lambda _{ji} $$ can be linearized on an abelian variety, i.e. are solvable by quadratures. We show that, merely by requiring that the solutions of the differential equations be single-valued functions of complex timet∈?, suffices to prove that (under a non-degeneracy assumption on the metric λ) the only such metrics are those which satisfy Manakov's conditions λ _ij =(b _i ?b _j ) (a _i ?a _j )^?1. The case of degenerate metrics is also analyzed. ForN=4, this provides a new and simpler proof of a result of Adler and van Moerbeke [3].