The contact problem for hollow and solid circular cylinders with a symmetrically fitted belt and stressfree faces is considered. Homogeneous solutions corresponding to zero stresses on the cylinder faces are obtained. The generalized orthogonality of homogeneous solutions is used to satisfy the modified boundary conditions. In the final analysis the problem is reduced to a system of integral equations in the functions describing the displacement of the outer and inner surfaces of the cylinders. These functions are sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result, are regularized by introducing small positive parameters [Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978] and, after reduction, have stable regularized solutions. Since the elements of the matrices of the system are given by poorly converging numerical series, an effective method of calculating the residues of these series is developed. Formulae for the distribution function of the contact pressure and the integral characteristic are obtained. Since the first formula contains a third-order derivative of the functional series, a numerical differentiation procedure is employed when using it [Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. A Student Textbook. Moscow: Vysshaya Shkola; 1976]. Examples of the analysis of a cylindrical belt are given.