Deformation quantization of integrable systems

In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra C in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of C with coefficients in A. In some particular case of the pair (A,C) we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra C is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.