Third order operator for the good Boussinesq equation on the circle

We consider a non-self-adjoint third order differential operator on R with real 1-periodic coefficients. The Lax equation for this operator is equivalent to the so-called good Boussinesq equation on the circle. The eigenvalues of the monodromy matrix constitute a 3-sheeted Riemann surface. Ramifications of this surface are invariant with respect to the Boussinesq flow. We determine high energy asymptotics of the ramifications.