Absence of local conserved quantity in the Heisenberg model with next-nearest-neighbor interaction

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.