  A third order self-adjoint differential operator with periodic boundary
conditions and an one-dimensional perturbation has been considered. For this
operator, we first show that the spectrum consists of simple eigenvalues and
finitely many eigenvalues of multiplicity two. Then the expressions of
eigenfunctions and resolvent are described. Finally, the inverse problems for
recovering all the components of the one-dimensional perturbation are solved.
In particular, we prove the Ambarzumyan-type theorem and show that the even or
odd potential can be reconstructed by three spectra.
