  This paper continues our recent studies on the algebraic Bethe ansatz for the
RTT-algebras of sp($2n$) and o($2n$) types. In these studies, we encountered
the RTT-algebras which we called An. The next step in our construction of the
Bethe vectors for the RTT-algebras of type sp($2n$) and o($2n$) is to find the
Bethe vectors for the RTT-algebras $\mathcal{A}_n$. This paper deals with the
construction of the Bethe vectors of the RTT-algebra An using the Bethe vectors
of the RTT-algebra $\mathcal{A}_{n-1}$.
