  We consider open spinning string solutions on an AdS_4 x S^2-brane (D5-brane)
in the bulk AdS_5 x S^5 background. By taking account of the breaking of
SO(6)_R to SO(3)_H x SO(3)_V due to the presence of the AdS-brane, the open
rotating string ansatz is discussed. We construct the elliptic folded/circular
open string solutions in the SU(2) and the SL(2) sectors, so that they satisfy
the appropriate boundary conditions. On the other hand, in the SU(2) sector of
the gauge theory, we compute the matrix of anomalous dimension of the defect
operator, which turns out to be the Hamiltonian of an open integrable spin
chain. Then we consider the coordinate Bethe ansatz with arbitrary number of
impurities, and compare the boundary condition of the Bethe wavefunction with
that of the corresponding open string solution. We also discuss the Bethe
ansatz for the open SL(2) spin chain with several supports from the string
theory side. Then, in both SU(2) and SL(2) sectors, we analyze the Bethe
equations in the thermodynamic limit and formulate the `doubling trick' on the
Riemann surface associated with the gauge theory.
