  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.
