  We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin
chain in the case when both boundary matrices can be brought to the
upper-triangular form. We define the Bethe vectors which yield the strikingly
simple expression for the off shell action of the transfer matrix, deriving the
spectrum and the corresponding Bethe equations. We explore further these
results by obtaining the off shell action of the generating function of the
Gaudin Hamiltonians on the Bethe vectors through the so-called quasi-classical
limit.
