Mc'Kean's transformation for 3-rd order operators

We consider a non-self-adjoint third order operator with 1-periodic coefficients. Integrating the Boussinesq equation on a circle requires solving the inverse spectral problem for this operator. In 1981, McKean introduced a transformation that reduces the spectral problem for this operator to the spectral problem for the Hill operator with a potential that depends analytically on the energy. In the present paper we are studying this transformation.