  We consider the third order operator with periodic coefficients on the real
line. This operator is used in the integration of the non-linear evolution
Boussinesq equation. For the minimal smoothness of the coefficients we prove
that: 1) the operator is self-adjoint and it is decomposable into the direct
integral, 2) the spectrum is absolutely continuous, fills the whole real axis,
and has multiplicity one or three, 3) the Lyapunov function, analytic on a
three-sheeted Riemann surface, is constructed and researched, 4) the spectrum
of multiplicity three is bounded and it is described in terms of some entire
function (the discriminant).
