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Use a common invariant operator domain on which the stated differentiations and commutator are valid. For a differentiable normalized eigenfunction, the Hellmann-Feynman calculation and self-adjointness giveThe Lax equation preserves the spectrum. Differentiating $Lf=\lambda f$ now gives $(L-\lambda )(f_t+Af)=0$, so $f_t+Af$ is in the same eigenspace, or is zero. For a nondegenerate eigenvalue it equals $c(t,\lambda )f$. Multiplying $f$ by $\exp [-\int c(t,\lambda )dt]$ produces $\hat{f}$ with ${\hat{f}}_t+A\hat{f}=0$ while preserving its eigenvalue.

Write $b=u_x+a_0$ and $d=u-\lambda +a_1$. The spatial eigenvalue equation gives ${\hat{f}}_{xx}=(u-\lambda )\hat{f}$, so $A\hat{f}=b\hat{f}+d{\hat{f}}_x$. For $F=(\hat{f},{\hat{f}}_x{)}^T$ the two first-order systems are thereforeThe second row of $V$ follows by differentiating the first time equation in $x$ and using the spatial equation once more. Conversely these matrix equations imply both scalar equations, so the formulations are equivalent.

Solve $\dot{Q}=-AQ$, $Q(0)=I$. Since $A^T=-A$, differentiation gives $Q^{T}Q=I$. To see why the Lax equation preserves the spectrum, direct differentiation givesThus $L(t)=Q(t)L(0)Q(t{)}^T$ is an orthogonal similarity, so all eigenvalues and all coefficients of its characteristic polynomial are first integrals.

A $2n$-dimensional Hamiltonian system is Liouville integrable when it has $n$ functionally independent first integrals, including the Hamiltonian, whose pairwise Poisson brackets vanish, on an open dense regular region.

For the three-particle periodic Toda lattice, take $p_i={\dot{q}}_i$. The proposed Hamiltonian haswhich are exactly the equations of motion. Differentiating the new variables givesThe transformation is not canonical. For examplerather than canonical coordinate brackets. Moreover $a_1{a}_2{a}_3=1/8$, so the map forgets the common translation of all the $q_i$ and is not an invertible coordinate change on the six-dimensional phase space.

Multiplication of the displayed matrices gives diagonal entries $2(a_i^2-a_{i-1}^2)$ and off-diagonal entries $(b_{i+1}-b_i)a_i$ in $LA-AL$. Therefore they form the required Lax pair. The three characteristic coefficients areThey are independent first integrals, as permitted without proof. Equivalently one may takeIndeed $C_1=-P/2$, $C_2=P^2/8-H/4$ and $C_3=1/4-J/8$. Their mutual Poisson commutation also follows directly: $P$ generates the common translation, and $\{H,J\}=0$ because $J$ is conserved.