The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.