Hidden symmetry of the Landau–Lifshitz equation, hierarchy of higher equations, and the dual equation for an asymmetric chiral field

A group-theoretical scheme is developed for constructing higher stationary Landau–Lifshitz equations as the Euler equations on the orbits of hidden symmetry algebra (the algebra of $su(2)$-valued functions on an elliptic curve). Duality between the Landau–Lifshitz equation and asymmetric $O(3)$-chiral field equation is established.