  We rigorously prove that the Heisenberg chain with next-nearest-neighbor
interaction, which is anticipated to be non-integrable, is indeed
non-integrable in the sense that this system has no nontrivial local conserved
quantity. Our result covers two important models, the Majundhar-Ghosh model and
the Shastry-Sutherland model, as special cases. These models are shown to be
non-integrable while have some solvable energy eigenstates.
