  We consider the self-adjoint third order operator with 1-periodic
coefficients on the real line. The spectrum of the operator is absolutely
continuous and covers the real line. We determine the high energy asymptotics
of the periodic, anti-periodic eigenvalues and of the branch points of the
Lyapunov function. Furthermore, in the case of small coefficients we show that
either whole spectrum has multiplicity one or the spectrum has multiplicity one
except for a small spectral nonempty interval with multiplicity three. In the
last case the asymptotics of this small interval is determined.
