A general approach is proposed in this paper to the solution of an axisymmetric problem about a circular crack of normal separation in an unbounded elastic body whose Poisson ratio depends in an arbitrary manner on the distance to the plane of the crack. By using the theory of the Hankel integral transform and associated dual integral equations, the problem mentioned is reduced to a symmetric Fredholm integral equation of the second kind, whose exact analytic solution is constructed for a specific law of Poisson ratio variation.