Towards the Proof of Complete Integrability of Quantum Elliptic Many-body Systems with Spin Degrees of Freedom

We consider the problem of finding integrals of motion for quantum elliptic Calogero–Moser systems with arbitrary number of particles extended by introducing spinexchange interaction. By direct calculation, after making certain ansatz, we found first two integrals — quite probably, lowest nontrivial members of the whole commutative ring. This result might be considered as the first step in constructing this ring of the operators which commute with the Hamiltonian of the model.