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

We consider a non-self-adjoint third order operator $(y^{\mathrm{\prime }\mathrm{\prime }}+py{)}^{\mathrm{\prime }}+p{y}^{\mathrm{\prime }}+qy$ with 1-periodic coefficients $p,q$. This operator is the L-operator in the Lax pair for the good Boussinesq equation on the circle. In 1981, McKean introduced a transformation that reduces the spectral problem for this operator to a spectral problem for the Hill operator with a potential that depends analytically on the energy. In the present paper we are studying this transformation.