  In this article we study the relation between the eigenstates of open
rational spin $\frac{1}{2}$ Heisenberg chains with different boundary
conditions. The focus lies on the relation between the spin chain with diagonal
boundary conditions and the spin chain with triangular boundary conditions as
well as the class of spin chains that can be brought to such form by certain
similarity transformations in the physical space. The boundary driven Symmetric
Simple Exclusion Process (open SSEP) belongs to the latter. We derive a
transformation that maps the eigenvectors of the diagonal spin chain to the
eigenvectors of the triangular chain. This transformation yields an essential
simplification for determining the states beyond half-filling. It allows to
first determine the eigenstates of the diagonal chain through the Bethe ansatz
on the fully excited reference state and subsequently map them to the
triangular chain for which only the vacuum serves as a reference state. In
particular the transformed reference state, i.e. the fully excited eigenstate
of the triangular chain, is presented at any length of the chain. It can be
mapped to the steady state of the open SSEP. This results in a concise
closed-form expression for the probabilities of particle distributions and
correlation functions in the steady state. Further, the complete set of
eigenstates of the Markov generator is expressed in terms of the eigenstates of
the diagonal open chain.
