Integrabilities of the long-range t-J models with twisted boundary conditions

The integrability of the one-dimensional long-range supersymmetric t-J model has previously been established for both open systems and those closed by periodic boundary conditions through explicit construction of its integrals of motion. Recently the system has been extended to include the effect of magnetic flux, which gives rise to a closed chain with twisted boundary conditions. While the t-J model with twisted boundary conditions has been solved for the ground state and full energy spectrum, proof of its integrability has so far been lacking. Tn this paper eve extend the proof of integrability of the long-range supersymmetric t-J model and its SU(m\n) generalization to include the case of twisted boundary conditions.