  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
$H^{-s}(\mathbb{T},\mathbb{C})$, $0 \le s < 1/2$, with small imaginary part and
prove analytic properties of the moment map, defined in terms of spectral data
of $L_u$.
