Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 33E / a / Solution

Since $L^T=L$ and $B^T=-B$,so the commutator is symmetric. If $Lv=\lambda v$ with $\Vert v\Vert =1$, symmetry permits a differentiable orthonormal eigenbasis locally, andThus a Lax pair evolution is isospectral.

For the harmonic oscillator,With the stated $L$, takeDirect calculation givesThe eigenvalues of $L$ areTheir constancy proves conservation of the oscillator energy $H={\lambda }_{+}^2/2$.