Nonstandard Bethe Ansatz equations for open O(N) spin chains

The double row transfer matrix of the open $O\left(N\right)$ spin chain is diagonalized and the Bethe Ansatz equations are also derived by the algebraic Bethe Ansatz method including the so far missing case when the residual symmetry is $O(2M+1)\times O(2N-2M-1)$. In this case the boundary breaks the "rank" of the $O\left(2N\right)$ symmetry leading to nonstandard Bethe Ansatz equations in which the number of Bethe roots is less than as it was in the periodic case. Therefore these cases are similar to soliton-nonpreserving reflections.