Inverse problem for the $L$-operator in the lax pair of the Boussinesq equation on the circle

We consider a third-order non-self-adjoint operator which is an $L$-operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that, in a neighborhood of zero, our mapping is analytic and one-to-one.