Inverse problem for the L-operator in the Lax pair of the Boussinesq's equation on the circle

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