On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus

We investigate the spectrum of the Lax operator $L_u$ of the Benjamin-Ono equation on the torus for complex valued potentials $u$ in the Sobolev space $H^{-s}(\mathbb{T},\mathbb{C})$, $0\le s<1\mathrm{/}2$, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of $L_u$.