Multicomponent generalization of the hierarchy of the Landau–Lifshitz equation

We construct a second-order $2N$-component integrable system (with arbitrary $N$) whose spectral parameter lies on a curve of genus $g=1+(N-3)2^{N-2}$. The odd-order flows admit $N$-component reductions, which for $N=3$ coincide with the odd-order flows of the hierarchy of the Landau–Lifshitz equation.