  An order four automorphism of a Lie algebra gives rise to an integrable
system discussed by Terng. We show that solutions of this system may be
identified with certain vertically harmonic twistor lifts of conformal maps of
surfaces in a Riemannian symmetric space.
  Specialising to 4-dimensional target, we find that surfaces with holomorphic
mean curvature in 4-dimensional spaces with constant sectional or holomorphic
sectional curvatures constitute an integrable system as do Hamiltonian
stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space
(this last being a result of Helein-Romon).
