  Motivated by Heilmann and Lieb's work, we discuss energy level crossings for
the one-dimensional Hubbard model through the Bethe ansatz, constructing
explicitly the degenerate eigenstates at the crossing points. After showing the
existence of solutions for the Lieb--Wu equations with one-down spin, we solve
them numerically and construct Bethe ansatz eigenstates. We thus verify all the
level crossings in the spectral flows observed by the numerical diagonalization
method with one down-spin. For each of the solutions we obtain its energy
spectral flow along the interaction parameter U. Then, we observe that some of
the energy level crossings can not be explained in terms of U-independent
symmetries. Dynamical symmetries of the Hubbard model are fundamental for
identifying each of the spectral lines at the level crossing points. We show
that the Bethe ansatz eigenstates which degenerate at the points have distinct
sets of eigenvalues of the higher conserved operators. We also show a twofold
permanent degeneracy in terms of the Bethe ansatz wavefunction.
