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The $(j,j+1)$ entry of $dotL$ is $dot{a}_j$. Comparing it with the result of part (i) givesFor $q_j=a_j^2$, this becomes $dot{q}_j=2q_j(q_{j+1}-q_{j-1})$, the Finite Volterra lattice.

Let $L$ be the symmetric zero-diagonal Jacobi matrix with $L_{j,j+1}=L_{j+1,j}=a_j$, and let the skew-symmetric matrix $B$ haveThen $\dot{L}=[B,L]$ is equivalent to the Finite Volterra lattice. The corresponding similarity flow makes the eigenvalues of $L$, the coefficients of its characteristic polynomial, and its trace invariants first integrals.