Non-Abelian Euler top

Abstract

In [1], [2] a general approach to constructing integrable non-commutative generalizations of a given integrable system with polynomial right-hand side was proposed. We apply it to finding non-commutative analogues of the Euler top. Consider the system of ODEs u′ = z1 vw, v′ = z2 uw, w′ = z3 uv, zi ∈ C, zi ̸= 0, (1) where ′ means the derivative with respect to t. The system (1) possesses the first integrals I1 = z3u − z1w and I2 = z3v − x2w. For any i and j, the system uτ = z1 vwI 1I j 2 , vτ = z2 uwI i 1I j 2 , wτ = z3 uvI i 1I j 2 (2)