  We show that the subregion entanglement Hamiltonians of excited eigenstates
of a quantum many-body system are approximately linear combinations of
subregionally (quasi)local approximate conserved quantities, with relative
commutation errors $\mathcal{O}\left(\frac{\text{subregion boundary
area}}{\text{subregion volume}}\right)$. By diagonalizing an entanglement
Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can
obtain these conserved quantities as the EHSM eigen-operators with nonzero
eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues
is cut off around the order of subregion volume, and some of their EHSM
eigen-operators can be rather nonlocal, although subregionally quasilocal. In
the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues
decay roughly in power law if the system is integrable, with the exponent
$s\approx 1$ ($s\approx 1.5\sim 2$) if the eigenstates are extended (many-body
localized). For fully chaotic systems, only two EHSM eigenvalues are
significantly nonzero, the eigen-operators of which correspond to the identity
and the subregion Hamiltonian.
