The integrability of the deformed quantum elliptic Calogero-Moser problem introduced by Chalykh, Feigin and Veselov is proven. Explicit recursive formulae for the integrals are found. For integer values of the parameter this implies the algebraic integrability of the systems.

Explicit algebraic relations between the quantum integrals of the elliptic CalogeroMoser quantum problems related to the root systems A 2 and B 2 are found.

Algebraic integrability of the elliptic Calogero-Moser quantum problem related to the deformed root systems A 2 (2) is proved. Explicit formulae for integrals are found.Following to [1] (see also [2] and [3]) we call a Schrödinger operatorintegrable if there exist n commuting differential operators L 1 = L, L 2 , . . . , L n with constant algebraically independent highest symbols P 1 ξ = ξ 2 , P 2 ξ , . . . , P n ξ , and algebraically integrable if there exists at least one more differential operator L n+1 , which commutes with the operators L i , i = 1, . . . , n, and whose highest symbol P n+1 ξ is also independent on x and takes different values on the solutions of the algebraic systemfor generic c i .The question how large is the class of the algebraically integrable Schrödinger operators is currently far from being understood, so any new example of such an operator is of substantial interest.